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this notes represent the written version of lectures i gave in mini - courses at universidade de braslia ( april 3 - 6 , 2013 ) , universidad central de venezuela ( may 23 - 27 , 2016 ) and at the villa de leyva summer school `` geometric , topological and algebraic methods for quantum field theory '' ( july 15 - 27 , 2013 ) . they were mainly intended as an introduction to some aspects of operator algebras , emphasizing the prominent role they play in quantum physics . as the audience consisted of students of both physics and mathematics at different stages of their studies , my choice was to focus on the most basic structures and examples , in the hope that a good grasp of these would motivate them to go deeper into the subject . now , something that to a physicist may appear as completely familiar ( as , say , an experimental set - up with polarizers , or the distinction between a classical and a quantum field ) to a mathematician may not . the same could be said of the proof that the spectrum of any element in a @xmath0-algebra is never empty : it is a standard result in analysis , but may look quite awkward to many physics students . therefore , the emphasis of these lecture notes will be on explaining _ why _ certain mathematical structures may be useful for the study of quantum theory . this will be illustrated by means of several examples that include a discussion of bipartite entanglement , algebraic and geometric aspects of quantum phase transitions in spin chains , quasi - free states in fermionic systems and applications to quantum field theory . our starting point will be based on the sharp distinction between _ classical probability _ and the probability theory inherent to quantum mechanics . this will provide a physical motivation to the various mathematical concepts we will be dealing with . after introducing the basic concepts about @xmath0-algebras , we will show how an algebraic approach to entanglement can lead to a resolution of certain discrepancies appearing when we deal with systems of identical particles . then we will focus on the study of certain specific models ( quantum ising and xy models ) that turn out to be ideal in order to illustrate how entanglement , geometry and the theory of car algebras - algebras used to model fermionic systems ] are interrelated . our first objective will be to understand what are the main structural differences between _ classical _ and _ quantum _ physics , when regarded as probability theories . this will , by the way , provide a physical motivation for some of the mathematical notions we will consider in the next sections . let us recall that classical physics is usually modeled on a configuration ( or phase ) space , with a dynamics governed by , _ e.g. _ , hamilton s principle . on the other hand , quantum mechanics is modeled on a hilbert space . so the first issue we want to explore is : why do we have to use quantum mechanics to describe the microscopic world ? indeed , simple experiments with light polarizers make it clear that there is no way we can describe certain phenomena using classical physics or , better said , classical _ probability_. let us then explore some elementary polarization phenomena , following the presentation in  @xcite . first let us recall that light is just made of electromagnetic waves , their behavior being governed by maxwell s equations : @xmath1 where @xmath2 ( resp . @xmath3 ) stands for the electric ( resp . magnetic ) field , @xmath4 for the charge density and @xmath5 for the current density . [ ex : intro1 ] show that maxwell s equations ( [ eq : intro - maxwell ] ) in vacuum ( @xmath6 ) lead to plane - wave solutions for the electromagnetic field propagating at a speed @xmath7 and such that @xmath2 and @xmath3 are always perpendicular to each other and to the propagation direction . what is the relation between the _ intensity _ of the wave ( defined as @xmath8 ) and the energy content of the fields ?    in simple terms , a _ polarizer _ is a filter that only allows the transmission of light waves which have a specific polarization angle . if we let unpolarized light go through an ideal polarizer , the intensity @xmath9 of the transmitted wave will be found to be half the intensity @xmath10 of the incident wave : @xmath11 ( malus law ) . after passing through the polarizer , the light is said to be linearly polarized . let us now suppose that we have a beam of linearly polarized light and we let it go through a second polarizer , such that its polarization axis has been rotated by an angle @xmath12 with respect to the axis of the first polarizer . then , experiment tells us that the intensity of the transmitted light will be @xmath13 . this is all fine if we are working with classical electromagnetic waves , which are described by maxwell s equations . in this case we just need to consider the projection of the field onto the direction singled out by the polarization axis . but we know that light is actually made of photons and , if their number is small , we are led to regard the @xmath14 term as a kind of `` expectation value '' .    in a very influential paper @xcite , einstein , podolski and rosen presented a criticism of quantum theory in what is now known as the _ epr paradox_. this led to the development of alternative , so - called hidden variable " theories that aimed at explaining physical phenomena using classical probability models . it was only until bell proved his famous inequalities , and aspect s experiments proved the former were violated , that the controversy could be resolved , showing that quantum theory provided the correct description of the phenomena . an experimental set - up , of the type studied by aspect , consists of a source ( ca atom ) located in the middle that emits simultaneously a pair of photons . one of them goes to the right , the other to the left . there are two detectors , one at each extreme . there is also a polarizer in - between each detector and the source , as depicted in fig . [ fig : intro - ca - atom ] .    , @xmath15 ) . there is a polarizer between the ca atom and each of the detectors . ] let us now consider the following two propositions : + @xmath16``left photon passes through ( is detected by @xmath17 ) when polarizer s angle is @xmath12 '' . + @xmath18``right photon passes through ( is detected by @xmath15 ) when polarizer s angle is @xmath19 '' . + then , what we learn from experiment is that the _ joint probability _ for both photons passing through the polarizers , thus being detected , is @xmath20 before exploring why classical probability is in conflict with this result , let us recall how the principles of quantum mechanics allow us to predict it . let @xmath21 denote a hilbert space which , for simplicity , will be considered to be finite dimensional . in quantum mechanics , probability distributions are obtained from state vectors @xmath22 or , more generally , by density matrices @xmath4 , that is , self - adjoint positive operators of trace one . on the other hand , observables are described by self - adjoint operators . let @xmath23 be such an observable . let @xmath24 denote its spectrum . we can then consider its spectral decomposition @xmath25 if the state of the system is given by a density matrix @xmath4 , the _ expectation value _ of @xmath23 is defined as @xmath26 in the case of a _ pure _ state , the density matrix is a rank - one projector , of the form @xmath27 , @xmath28 , and so ( [ eq : intro < a > ] ) reduces to @xmath29 . in this way we obtain a probability distribution over @xmath30 , with @xmath31 in fact , it follows from ( [ eq : intro - speca ] ) and ( [ eq : intro < a > ] ) that @xmath32 and @xmath33 . let us use this to give a mathematical description of the experiment described above . the polarization state of a photon can be described using a 2-dimensional hilbert space . let @xmath34 denote an orthonormal basis , that can be used to describe , say , horizontal and vertical polarization states . since we are considering a system consisting of two photons , the hilbert space of the system can be taken to be @xmath35 ( as the photons are supposed to be far from each other at the time of detection , the symmetrization postulate can be ignored ) . the 2-photon state , as produced by the emission from an excited state of the ca atom , can be described by the following state vector : @xmath36 we can use @xmath37 , defined as the _ projector _ onto @xmath38 . since we have two polarizers , we introduce the following projectors ( _ cf . _ fig . [ fig : intro - ca - atom ] ) : @xmath39 it is important to remark that these two operators correspond to compatible observables , in the sense that the measurement of one of them does not affect the result of the other , _ i.e. _ , they are commuting observables : @xmath40=0.\ ] ]    show that @xmath41 also show that , for the _ joint _ measurement of the two polarization states , one obtains @xmath42    the previous exercise shows that the predictions of quantum mechanics for this type of experiment are in accordance with what is actually measured in the laboratory . in spite of its extreme simplicity , there is an intriguing feature of this result : by appropriately arranging the polarization angles @xmath19 and @xmath12 , we can obtain a _ total anticorrelation _ for the joint measurements . as we will see , this is due to the fact that the state ( [ eq : intro-2-photon - state ] ) is an _ entangled state_. that these kind of correlations can not be obtained from a ( local , realistic ) classical theory is at the core of the original epr controversy . for an interesting discussion of these issues in the context of an actual experimental situation , we recommend  @xcite . our immediate aim will therefore be to understand where exactly classical probability fails at describing the results of such experiments . for this purpose we will assume an approach to ( classical ) probability based on propositions and degrees of plausibility , following jaynes  @xcite .    in this setting , the objects to be considered are the following :    * a set of propositions : @xmath43 , each one of which can take on ( only ) two values : true or false . * logical operations , that can be performed on the set of propositions : * * conjunction , or logical product ( and ) : @xmath44 . it is true if and only if both @xmath23 and @xmath3 true . * * disjunction , or logical sum ( or ) : @xmath45 . it is true if at least one of them true . * * negation ( not ) : @xmath46 , with opposite truth value as @xmath23 . these logical operations are assumed to satisfy the defining rules of a _ boolean algebra _ :    * idempotency : @xmath47 * commutativity : @xmath48 * associativity : * * @xmath49 * * @xmath50 * distributivity : * * @xmath51 * * @xmath52 * duality ( de morgan s laws ) : @xmath53    show that the proposition `` @xmath54 '' is equivalent to `` @xmath55 '' . show that ( d2 ) follows from the other rules . now , given a proposition @xmath23 , a probability ( or `` degree of plausibility '' , _ cf . _  @xcite ) @xmath56 can be assigned to it , under the following basic assumptions :    * let @xmath57 denote the probability , or degree of plausibility , of a given proposition @xmath23 . then we assume that : * * @xmath58 , for any logical function @xmath59 of the propositions @xmath60 * * using the notation @xmath61 for the conditional probability that @xmath23 is true , given @xmath3 is true , we assume that @xmath62 whenever @xmath63 more plausible than @xmath64 . * given a proposition @xmath65 , let us suppose that we improve our state of knowledge , obtaining a new proposition @xmath66 . if as a result @xmath23 becomes _ more _ plausible , _ i.e. _ @xmath67 , but @xmath68 , then @xmath69 should hold ( this is dubbed the _ common sense _ assumption by jaynes  @xcite ) .    from i and ii above we may obtain , under very general assumptions ( like consistency ) , the following two basic rules :    * product rule:@xmath70 * sum rule : @xmath71    the reader is invited to consult jaynes for a comprehensive exposition of this point of view , including many illustrative derivations and examples . now , in order to return to our experiment , let us define the following `` coincidence '' function @xmath59 , for two given propositions @xmath23 and @xmath3 : @xmath72    given 4 propositions @xmath73 , use the product and sum rules to show that the following proposition is true : @xmath74    since @xmath75 whenever @xmath54 holds ( product rule ) we obtain , from the previous exercise , + `` bell s inequality '' : @xmath76 referring back to figure [ fig : intro - ca - atom ] , let us consider the following propositions ( @xmath77 ) : + @xmath78 `` left photon passes through ( is detected by @xmath17 ) when polarizer s angle is @xmath79 '' . + @xmath80 `` right photon passes through ( is detected by @xmath15 ) when polarizer s angle is @xmath81 '' . the inequality ( [ eq : intro - bell - chsh ] ) should hold true for all choices of @xmath82 . but this then implies : @xmath83 for the choice @xmath84 , this means : @xmath85    @xmath86 @xmath87 this contradiction means that the assumptions we have considered above ( which are at the basis of classical probability ) do not apply in the quantum realm . so , definitely , quantum theory leads to a very different type of probability theory . in the next section we will explore some of the more notorious differences between classical and quantum probabilities . until now we have avoided any mention of propositions in terms of set theory , which is the basis of the kolmogorov axiomatics . in the finite dimensional case , at least , they turn out to give equivalent structures . let us then consider the main properties of a ( classical ) probability theory , formulated in terms of set theory and , for simplicity , in the finite dimensional context . let us consider a finite set @xmath88 , regarded here as the _ sample space_. the _ event space _ @xmath89 is a certain collection of subsets of @xmath90 , that must contain the empty set and be closed under complements and unions . the set of events forms a boolean algebra , under the standard set theoretic operations :    * and : @xmath91 , * or : @xmath92 , * not : @xmath93 .    a probability distribution is then defined as a map @xmath94\subset \mathds r$ ] , such that    * @xmath95 @xmath96 , * @xmath97 for @xmath98 pairwise disjoint . notice that the space of probability distributions is a convex set , for if @xmath99 and @xmath100 are two probability distributions , then @xmath101 is again a probability distribution , provided @xmath102 . but not only is the space of probability distributions a convex set , it is a special kind of convex set : a _ simplex_. to see this , all we have to do is to consider the following `` extremal '' distributions , @xmath103 , defined by @xmath104 it is clear that any @xmath105 can be written as a _ unique _ convex combination of these extremal distributions , as we have @xmath106 with @xmath107 . a random variable ( or `` observable '' ) is a function @xmath108 . we call the space of all such functions @xmath109 . notice that @xmath109 forms a commutative algebra . summarizing , in the finite - dimensional case we have the following structure :    * the event space ( @xmath110 ) forms a boolean algebra . * the space of probability distributions is _ convex _ and furthermore has the structure of a simplex : every probability distribution @xmath105 can be uniquely written as convex combination of  extremal " distributions . * the space of observables ( random variables ) has the structure of a commutative algebra .    in cases where the sample space is not a finite set ( like in classical mechanics ) we need a suitable generalization of the above definitions . this is afforded by measure theory , in the following way : sample and event spaces are now replaced by a measurable space @xmath111 , where @xmath89 is a suitable @xmath112-algebra of @xmath90 . probability distributions are then defined as _ normalized , positive measures_. in particular , we can now handle countable additivity : @xmath113 finally , the space of observables is still a commutative algebra , for the product and sum of two measurable functions is a measurable function . a similar remark applies to the convex structure of the space of probability distributions . how do these structures show up in classical physics ? in classical mechanics , for example , the dynamics of a system can be described in terms of canonical variables `` position '' ( @xmath114 ) and `` momentum '' ( @xmath115 ) , that give rise to the _ phase space _ of the system . it follows from hamilton s variational principle that if @xmath116 is the hamiltonian of the system , the canonical variables will evolve along solutions to hamilton s equations : @xmath117 usually the phase space is the cotangent bundle @xmath118 of some configuration space @xmath119 . the canonical variables @xmath120 are then local coordinates on @xmath118 and the _ observables _ are smooth functions on phase space : @xmath121 . since time evolution is given by hamilton s equations or , at the level of observables by @xmath122 , with @xmath123 denoting the poisson bracket , one would think that there is no much space for a probabilistic model here since , given a set of initial conditions , by solving the equations of motion we are able , in principle , to predict the position and momenta of all the particles . but , as we learn from classical statistical physics , for @xmath124 ( @xmath125 ) we certainly need a statistical approach ! therefore we are forced to introduce probability distributions , expressed in terms of probability densities @xmath126 , so that the average value of an observable @xmath127 is given by @xmath128 where @xmath129 is the liouville measure on phase space . for instance , for the _ canonical ensemble _ ( used to describe a subsystem embedded in a thermal bath at temperature @xmath130 ) , we have @xmath131 . now we can return to the case of a few point particles and interpret the state of the system at a given time , usually defined as just a point @xmath132 in phase space , as an extremal probability distribution ( a `` pure state '' ) , for which the probability density is just a dirac delta distribution : @xmath133 . in any case , we see that classical physics can be regarded as a special type of probability theory , where the three properties of classical probability highlighted above still hold , _ i.e. _ , every classical system of point particles can be understood as a probability theory , in the kolmogorov sense .    what about quantum mechanics ? as discussed above , probability distributions are obtained from state vectors @xmath134 in a hilbert space @xmath21 , or more generally from density matrices , @xmath4 , whereas observables are described by self - adjoint operators . consider the spectral decomposition of a self - adjoint operator @xmath23 , as in ( [ eq : intro - speca ] ) . assuming a non - degenerate , discrete spectrum , we notice that the projectors @xmath135 , having as spectrum the set @xmath136 , can be regarded as `` indicators of events '' . now , there is a correspondence between projections @xmath137 and subspaces @xmath138 . the partial order that we naturally obtain by inclusion then gives rise to the structure of an `` orthocomplemented lattice of proposition '' , where the corresponding operations are defined as @xmath139 an important feature of this system is that it _ does not _ give rise to a boolean algebra structure . the reason for this is that the orthogonal complement is not the only possibility for a complement in this lattice . this in turn implies the breakdown of the distributive law , which is part of the definition of a boolean algebra . as in the classical case , the state space ( here the space of density matrices ) is a convex space . but , in contrast to the classical case , this space is not a simplex . a consequence of this is that the representation of a density matrix as a convex sum of pure states is ( highly ) non - unique . finally , in quantum theory the space of observables forms a non - commutative algebra , in contrast to the classical case , where the algebra is commutative . it is now clear that , in principle , both approaches ( classical and quantum ) can be considered in order to describe physical phenomena in probabilistic terms . in fact , epr - like arguments are in favor of a `` local - realistic '' point of view according to which even quantum phenomena should be explained in terms of classical probability . that this is not the case is proved by bell s inequalities and their violation , experimentally verified by aspect in the 80 s . so quantum theory can be regarded as a kind of `` non - commutative '' probability theory . one of the points of these lectures is that , when formulated in the language of operator algebras , both quantum and classical physics can be described in a unified way . we will also take advantage of this formulation to discuss a similar phenomenon , occurring in topology and geometry : in non - commutative geometry  @xcite the generalization of topological / geometric notions to the non - commutative setting has been mainly achieved by first expressing them in algebraic terms and then realizing that dropping the commutativity assumption allows for vast generalizations . let @xmath21 be a separable hilbert space . recall that , given a linear operator @xmath140 , @xmath130 is said to be bounded if there is some @xmath141 such that @xmath142 for all @xmath143 in @xmath21 , where the norm is the one induced by the inner product : @xmath144 . if @xmath130 is a bounded operator , we define its norm as follows : @xmath145{\mbox{sup } } { } \frac{\|t(x)\|}{\|x\|}.\ ] ] one then checks that for @xmath130 and @xmath146 bounded the inequalities @xmath147 are satisfied . from the completeness of @xmath21 it follows that the space @xmath148 is a complete normed space . it is also an algebra and has an involution `` @xmath149 '' given by the adjoint : @xmath150 . we thus may call @xmath151 the `` @xmath149-algebra of bounded operators on @xmath21 ''    show that for @xmath152 we have : @xmath153    we now abstract these notions and make them independent of any underlying hilbert space . as we will see below , a lot will be gained from this , since we will then be able to study _ representations _ of the ( abstract ) operator algebras , and the equivalence / inequivalence of these representations will have a deep physical meaning . a _ banach space _ is a normed vector space @xmath154 which is complete with respect to the ( metric induced by ) @xmath155 . a _ banach algebra _ is a banach space @xmath156 which is also an algebra , with the property that @xmath157 ( which in turn implies that multiplication is a continuous operation ) . an _ involution _ on a ( complex ) algebra @xmath158 is a map @xmath159 such that , for any @xmath160 and @xmath161 :    * @xmath162 * @xmath163 * @xmath164    notice that this is , basically , an abstraction of the adjoint operation on @xmath151 .    a _ @xmath0-algebra _ is a banach @xmath149-algebra @xmath165 with the fundamental property @xmath166    [ example : hausdorff ] let @xmath167 be a compact , hausdorff topological space and set @xmath168 , the space of continuous complex functions on @xmath167 and , for @xmath169 , set @xmath170 , and @xmath171{\mbox{sup } } { } |f(x)|$ ] . then @xmath172 is a @xmath0-algebra .    let @xmath21 be a hilbert space . then , as we expect , @xmath173 is a @xmath0-algebra , where the norm is the operator norm and the involution is given by the adjoint operation . later we will see that the list of examples of @xmath0-algebras is basically exhausted by the previous two examples , a remarkable fact . a motivation to work with algebras of bounded operators comes from physics since , as can be shown , the _ canonical commutation relations _ ( ccr ) @xmath174 = i\hbar \mathds 1\ ] ] can not be implemented by means of bounded operators @xmath175 and @xmath176 . then , although these commutation relations have a very clear meaning from the physical point of view , mathematically they correspond to _ unbounded _ operators and hence issues like self - adjointness , domains , etc . come into play , which make them more difficult to deal with . one can , nevertheless , replace the ccr by their exponentiated ( or weyl ) form , as follows . first we define operators @xmath177 and @xmath178 , for @xmath179 , acting on wave functions as follows : @xmath180 by stone s theorem , it follows that @xmath181 and @xmath182 . [ ex : weyl - ccr ] show that the operators @xmath177 and @xmath178 satisfy the following commutation relations : @xmath183    as we will see later on , these operators give rise to a @xmath0-algebra . in fact , we will see that to any symplectic vector space we can canonically associate a @xmath0-algebra ( its weyl @xmath0-algebra ) . the unitary representations of these algebras , in the case of infinite dimensional symplectic vector spaces , play a prominent role in the study of quantum field theory on curved spacetimes .    consider a ( complex , involutive , with unit @xmath184 ) algebra @xmath158 generated by elements @xmath185 , subject to the following _ canonical anticommutation relations ( car ) _ : @xmath186 if we want to define a @xmath0 norm on this algebra , it has to be such that the @xmath0-property ( [ eq : basics - cstar - ppty ] ) is satisfied . but from ( [ eq : basics - car1 ] ) we obtain @xmath187 , which along with ( [ eq : basics - cstar - ppty ] ) implies @xmath188 . the only option we have , then , is to define @xmath189 for every @xmath190 . [ ex : car - algebra - finite - dim ] generalize the previous example to the case of a finite dimensional hilbert space @xmath191 , in the following way . let @xmath192 denote the inner product . for any pair of vectors , @xmath193 , consider generators @xmath194 satisfying the following relations ( car ) : @xmath195 where @xmath196 . how must @xmath197 be defined in this case ?    the ( mathematical ) notion of _ state _ for a @xmath0-algebra is also very close to the notion of quantum state . to appreciate this , consider a physical system modeled by the algebra @xmath198 of bounded operators on a hilbert space @xmath21 . as mentioned in the previous section , in physics we distinguish between two kinds of states :    * pure states : these are described by normalized vectors @xmath28 ( actually , by rays on hilbert space ) . if @xmath23 is an observable , then its expectation value is defined as @xmath199 * mixed states : they are described by operators @xmath200 such that @xmath201 , @xmath202 . the expectation value is given , in this case , by @xmath203    show that the condition for a mixed state ( described by a density matrix @xmath4 ) to be pure is @xmath204 . thus , all states can be described by density matrices . the space of all density matrices is naturally a convex space . the extremal elements then turn out to be the pure ones . check this assertion in the simple case @xmath205 . we therefore see that a state can be regarded as a mapping @xmath206 from the observable algebra to the complex numbers . it must be possible , then , to express the properties @xmath207 and @xmath202 only in terms of this mapping . this leads us to the general definition of state for a @xmath0-algebra . [ def : basics - state ] let @xmath158 denote a ( unital ) @xmath0-algebra , with unit @xmath184 . a positive linear functional @xmath208 such that @xmath209 is called a _ state_. we will denote the set of all states on @xmath158 with @xmath210 .    in this context , positivity of @xmath211 means that @xmath212 for all @xmath213 . if @xmath158 is not unital , we can replace the normalization condition by the condition @xmath214 . for unital algebras , both definitions coincide .    having in mind the example of @xmath151 , it then makes sense to have a notion of spectrum for an abstract @xmath0-algebra . this leads , in turn , to the spectral theorem for @xmath0-algebras  @xcite . let @xmath215 be an element of a unital @xmath0-algebra @xmath158 . then its _ spectrum _ is defined as the set @xmath216    let @xmath215 be an element of a unital @xmath0-algebra @xmath158 . then its _ spectral radius _ is defined as @xmath217    show that the space @xmath218 of @xmath219 matrices with complex entries is a @xmath0-algebra with respect to the norm defined as @xmath220 where @xmath221 is the eigenvalue of @xmath222 with the largest absolute value . show also that this norm coincides with the operator norm . the previous exercise illustrates the remarkable fact that , for a @xmath0-algebra ( @xmath223 ) , the norm of any element normal @xmath213 coincides with its spectral radius : @xmath224 this is an important fact , as it links topological properties of the algebra with algebraic ones . in particular , it implies that the norm in a @xmath0-algebra is _ unique_.    the following two properties will be very useful :    [ ex : cauchy - schwarz ] let @xmath225 be a positive linear functional on a @xmath0-algebra . prove that    * @xmath226 . * @xmath227 ( `` cauchy - schwarz inequality '' )    hint : consider @xmath228 as a quadratic form on @xmath229 . as an alternative , you may try deriving these relations just by thinking of @xmath230 as `` @xmath231 '' . another consequence of positivity is continuity :    let @xmath232 be a positive linear functional on a unital @xmath0-algebra @xmath158 . then @xmath232 is a continuous linear functional , and @xmath233 . if @xmath215 is positive ( @xmath234 and @xmath235 ) then we have , from the spectral radius formula , that @xmath236 $ ] . this means that @xmath237 for all @xmath238 . this can be restated as follows . define @xmath239 then @xmath240 and @xmath241 . but then it follows ( from the continuous functional calculus ) that @xmath242 , so that @xmath243 or , in other words , that @xmath244 is a positive operator . since @xmath232 is positive and linear , this implies @xmath245 . now , for an arbitrary @xmath246 , apply the cauchy - schwarz inequality from exercise [ ex : cauchy - schwarz ] with @xmath247 and @xmath248 , and use the @xmath0 property of the norm to obtain @xmath249 . it follows that @xmath232 is continuous , with @xmath250 . but we also have @xmath251 so that @xmath233 . thus , for unital @xmath0-algebras we conclude that the set of states over @xmath158 is a convex subset of the continuous dual @xmath252 of @xmath158 . we gather from all this that it makes sense to formulate quantum physics in terms of @xmath0-algebras . the general philosophy will be to describe a given physical system in terms of its `` algebra of observables '' , which will be here taken to be a @xmath0-algebra @xmath253 . then we will consider the following two `` dual '' notions :    * a _ quantum state _ will be defined as a state ( in the sense of definition [ def : basics - state ] ) on the algebra @xmath158 . * an _ observable _ will be an element of @xmath158 . furthermore , the `` pairing '' between an observable @xmath215 and a state @xmath211 will be given the physical interpretation of an _ expectation value _ , and we will write @xmath254 regarding the distinction between pure and mixed states , we can now call a state _ pure _ if it can not be written as a convex combination of other states ; otherwise we will call it a _ mixed _ state .      before discussing physical applications of the notions introduced in the previous section , we will take the opportunity to briefly review the characterization of @xmath0-algebras due to gelfand , naimark and segal . the characterization of commutative @xmath0-algebras ( the gelfand - naimark theorem ) is of fundamental importance , in particular because it leads to the notion of a _ `` noncommutative topological space '' _  @xcite . also , the so - called gns - construction discussed below is relevant not only because of its role in the characterization problem for noncommutative @xmath0-algebras , but also because of its striking consequences for the study of quantum systems ( equilibrium states , symmetry breaking , inequivalent vacua , among others )  @xcite .    a _ character _ of a @xmath0-algebra @xmath158 is a @xmath149-homomorphism @xmath255 . let us denote with @xmath256 the set of all characters of @xmath158 . now , it is easy to see that if @xmath257 is a character , then @xmath258 . this , together with the spectral radius formula ( [ eq : spectral - radius - formula ] ) , implies that @xmath259 , so that @xmath260 , where @xmath261 is the unit ball in the dual space . since ( by the banach - alaoglu theorem  @xcite ) this ball is compact in the weak-@xmath149 topology ( _ i.e. _ the one induced by the family of seminorms on @xmath252 : @xmath262 , @xmath246 ) then @xmath256 becomes a _ compact topological space _ in the subspace topology inherited from @xmath252 . this is quite interesting in view of example [ example : hausdorff ] as this is providing the converse statement . in fact , for a commutative , unital @xmath0-algebra @xmath158 , we define the _ gelfand transform _ @xmath263 as the map @xmath264 where @xmath265 . [ thm : gelfand - naimark]the gelfand transform is an isometric @xmath149-isomorphism @xmath266    rather than reviewing the proof of this theorem , we will consider some basic examples that should provide a good intuition about the correspondence between commutative @xmath0-algebras and ( locally compact , hausdorff ) topological spaces . consider the unital , commutative @xmath0-algebra @xmath158 generated by a unitary element @xmath267 ( _ i.e. _ @xmath268 ) , with norm fixed by the condition  @xcite that @xmath269 for all @xmath270 . theorem [ thm : gelfand - naimark ] states that this algebra must be the function algebra of a topological space which , as a set , is precisely the set @xmath256 of all characters of @xmath271 . since the algebra is generated by @xmath267 , a character @xmath257 is fixed by its value on @xmath267 . but @xmath272 , and so every character is of the form @xmath273 , _ i.e. _ @xmath274 . in order to see that actually @xmath275 , we proceed as in  @xcite by noticing that from example [ example : hausdorff ] and from the uniqueness of the norm , it follows that @xmath276 , so that we must have @xmath277{\mbox{sup } } { } |1+e^{-i\alpha}\mu(u)|=2.\ ] ] but this can only happen if for every @xmath270 there is a character @xmath257 with @xmath278 . the 2-sphere @xmath279 can also be easily characterized in terms of its algebra of continuous functions . in this case , the algebraic structure is well - known , as @xmath280 is generated by the spherical harmonics @xmath281 , their product being given as in the usual clebsch - gordan decomposition . more details on how @xmath279 can be obtained as the character space of such an algebra ( this involves an appropriate definition of the norm , as in the previous example ) can be found in  @xcite . take @xmath282 as above ( with the sup norm ) and consider the subalgebra @xmath283 consisting of all even functions on @xmath158 , that is , functions @xmath59 such that @xmath284 . then @xmath285 . this fact has been exploited in studies of the spin - statistics connection in quantum mechanics  @xcite . the isomorphism provided by theorem [ thm : gelfand - naimark ] is actually an equivalence between the categories of commutative @xmath0-algebras and locally compact , hausdorff topological spaces . this means , among other things , that all information regarding the topology of @xmath167 is encoded in the algebra @xmath286 : an open set in @xmath167 can be equivalently described in terms of a ideal in @xmath286 , closed sets are described by quotient algebras , metrizability of @xmath167 amounts to separability of @xmath286 , and so on  @xcite . this is the reason why the gelfand - naimark theorem is considered one of the main sources for the development of noncommutative geometry  @xcite . now we turn to the gns ( gelfand - naimark - segal ) construction . let @xmath158 be a @xmath0-algebra , and @xmath211 a state thereon . the gns construction furnishes a representation of @xmath158 on some ( @xmath211-dependent ) hilbert space . the basic idea is to use multiplication in @xmath158 in order to obtain a linear action of @xmath158 on a vector space . so we begin by regarding @xmath158 as a vector space @xmath287 ( _ i.e. _ we just `` forget '' multiplication ) . even though the underlying spaces are equal , it will be convenient to distinguish the _ algebra _ , with elements @xmath213 , from the underlying _ vector space _ , with elements @xmath288 . recalling the cauchy - schwarz inequality ( exercise [ ex : cauchy - schwarz ] ) , we realize that it makes sense to introduce the following sesquilinear form on @xmath289 : @xmath290 this is `` almost '' an inner product , as it may happen that @xmath291 for some @xmath292 . but ignoring that fact for a moment , we notice that the product on @xmath158 can be used to make @xmath158 act as a linear operator on @xmath289 : @xmath293 now , in order to have a hilbert space representation , we have to `` fix '' the problem with ( [ eq : w(a*b ) ] ) . this is done as follows . define @xmath294 , and show that it is a closed left - ideal of @xmath158 . we therefore obtain a hilbert space @xmath295 given by the completion of @xmath296 with respect to the inner product @xmath297 | [ b]\rangle_\omega:= \omega ( a^*b)$ ] , where we use @xmath298\rangle$ ] to denote the equivalence class of @xmath215 in ( the completion of ) @xmath299 . with this we have obtained a @xmath149-representation of @xmath158 by bounded operators acting on @xmath295 : @xmath300 where @xmath301\rangle:= |[ab]\rangle$ ] . obtain the following inequality : @xmath302    although the representation @xmath303 is a @xmath149-homomorphism , it is not isometric . but it can be shown that for every @xmath213 there is a state @xmath211 such that @xmath304 . then , if we consider the direct sum @xmath305 where in each sequence @xmath306 only countable many elements are different from zero . the representations @xmath303 then give rise to a representation @xmath307 on @xmath21 for which @xmath308 . the details of this construction can be found _ e.g. _ in  @xcite . the importance of this result is that _ any _ @xmath0-algebra is isometrically isomorphic to the @xmath0-algebra of bounded operators in some hilbert space .      in this section , our aim will be to study certain quantum correlations that arise as a result of _ entanglement_. this is a vast subject , and here we will only consider elementary examples , corresponding to bipartite systems . although very simple from the mathematical point of view , these examples already contain the essence that will allow us to distinguish between classical and genuinely quantum correlations . we will also discuss a recent application of the algebraic formalism to the study of entanglement for systems of identical particles  @xcite , a topic of current interest for quantum information , condensed matter , atomic physics , and quantum optics . we thus start by considering a composite system that is the result of coupling two subsystems 1 and 2 . we will first consider quantum systems described directly in terms of hilbert spaces . but , as will become clear , the algebraic approach based only on observable algebras and states will allow us to reformulate these ideas in more generality . this will be important for the applications to systems of identical particles . let @xmath309 and @xmath310 denote the corresponding ( finite dimensional , for simplicity ) hilbert spaces , and assume that the subsystems are described in terms of observable algebras @xmath311 , @xmath312 . subsystems 1 and 2 are then coupled to form a composite system that will be described by @xmath313 . let now @xmath314 be a state of the composite system . consider `` partial '' , or `` local '' measurements performed on each subsystem . this leads us to consider the restriction of @xmath211 to @xmath315 , @xmath312 : @xmath316    in the present context of a bipartite composite system , if @xmath211 is a pure state on @xmath158 , we say it is a _ separable _ state if @xmath317 for all @xmath318 , @xmath319 . a ( pure ) state that is not separable in the above sense is called an _ entangled _ state . let @xmath320 , with basis @xmath321 and @xmath322 , with basis @xmath323 . let @xmath324 and @xmath325 . the composite system is then described by the matrix algebra @xmath326 . pick now a vector state @xmath327 and set @xmath328 then , the following three conditions are equivalent :    * the vector state @xmath329 is of the form @xmath330 . * @xmath331 , ( @xmath312 ) where @xmath332 stands for the _ von neumann entropy _ of the reduced density matrix @xmath333 ( see definitions below ) . * the state @xmath334 is separable .    to see where these equivalences come from , let us expand @xmath329 as @xmath335 then @xmath336 , so we may perform a singular value decomposition and write it in the form @xmath337 , with @xmath338 a diagonal matrix in @xmath339 , @xmath340 and @xmath341 . the elements of @xmath338 are @xmath342 , where the @xmath343 , called the  schmidt coefficients " , are the eigenvalues of @xmath222 . if we define vectors @xmath344 from this expression , it is clear that ( @xmath345 ) holds precisely when there is only one non - vanishing schmidt coefficient . furthermore , from @xmath346 we obtain @xmath347 , with @xmath348 we can , therefore , regard @xmath349 as a probability distribution , and compute its ( shannon ) entropy , defined as @xmath350 notice that this function vanishes precisely when @xmath329 is a separable state . we can relate it to the von neumann entropy of the restricted states @xmath351 as follows . [ ex : partial - trace ] consider the restriction @xmath351 of @xmath334 to @xmath352 ( @xmath312 ) . find density matrices @xmath333 ( `` reduced density matrices '' ) acting on @xmath353 and such that @xmath354 and @xmath355 . the _ von neumann entropy _ of a density matrix @xmath4 is defined as @xmath356 show that @xmath357 and @xmath358 coincide and are exactly equal to @xmath359 from ( [ eq : h(lambda ) ] ) . we therefore consider the quantity @xmath332 ( @xmath312 ) as a _ measure _ of the `` amount of entanglement '' of the state @xmath334 . in order to really appreciate the meaning of this assertion , we have to understand the ( very surprising , and interesting ) features of quantum correlations . this can be considered as one of the starting points of quantum information theory . in quantum information theory , there are orderly ways to pose and study these type of problems where , for instance , in the ( pure ) bipartite case the von neumann entropy of the reduced density matrix can be obtained as ( basically ) the unique entanglement measure of the state @xmath329 , when entanglement is regarded as a physical resource and defined in operational terms  @xcite . the general problem , that of multipartite entanglement , is much more complicated and involves many mathematically as well as physically interesting problems . however , our interest here will be restricted to the ( pure ) bipartite case . this will be enough to illustrate one of our points , which is that the algebraic framework leads to a unified description of physical systems , no matter whether they are classical or quantum . also , the `` detachment '' from an _ a priori _ given hilbert space in the quantum case allows for a more clear understanding of phenomena . as it turns out , the noncommutativity of the observable algebra can be seen as the main source of ( truly quantum ) correlations . this fits nicely with the relation between commutative and noncommutative spaces , as hinted above . the next example illustrates the previous remarks . let us consider the composition of two classical systems . by this we mean that the `` coupling '' of two systems described by ( commutative ) observable algebras of the form @xmath360 and @xmath361 is effected by the tensor product @xmath362 . tensor products of @xmath0-algebras have to be treated carefully , but assuming we have defined the appropriate tensor product , let us assume that @xmath363 ( a detailed discussion about tensor products can be found in  @xcite ) . hence , the composite system will be assumed to be described by the observable algebra @xmath364 . consider now a state @xmath211 on @xmath158 . it is not difficult to see that it must be given by a probability distribution @xmath365 in such a way that for @xmath366 , @xmath367 restriction to @xmath368 or @xmath369 in this case leads to marginal distributions @xmath370 and @xmath371 , which in turn define states @xmath372 on @xmath373 . from the above definition we have that @xmath211 is separable if , and only if , @xmath374 but from the properties of the gelfand transform it follows that any pure state on @xmath375 is of the form @xmath376 , _ i.e. _ , they are given by point measures : @xmath377 . it follows immediately that @xmath378 . this illustrates the fact that entanglement entails purely quantum correlations , as all pure states of a classical composite ( bipartite ) system are separable . the previous example was formulated in rather loosely terms . provide the necessary details to turn it into a rigorous proof that works for locally compact spaces . [ example : psi_lambda ] on @xmath379 consider the following family of states : @xmath380 and the following types of observables , acting on @xmath381 : @xmath382 with @xmath383 a _ unit _ vector in @xmath384 and @xmath385 ( pauli matrices ) . straightforward computations give @xmath386 and @xmath387 whereas for @xmath388 we obtain separable states , for @xmath389 we have entangled states . in particular , for @xmath390 we obtain a maximally entangled state ( bell state ) , for which @xmath391 in this case , the marginal distributions give @xmath392 whereas for the joint probabilities we obtain ( @xmath393 ) : @xmath394 in particular , for the choice @xmath395 , we obtain _ total anticorrelation_.    what is so special about the previous example ? consider the clauser - horne - shimony - holt ( chsh ) inequality  @xcite : @xmath396 where @xmath397 denotes the expectation value of an observable @xmath398 , and where for the involved observables it is assumed that @xmath399 . the inequality can be obtained if we assume some underlying ( classical ) probability space @xmath400 , for which the correlations above take the form @xmath401 or , stated in terms of algebras of observables :    let @xmath158 be a _ commutative _ @xmath0-algebra , @xmath211 a pure state thereon , and @xmath402 elements of @xmath158 , all of them with norm less or equal to one . show that @xmath403    but , in contrast to this , in example [ example : psi_lambda ] ( with @xmath404 ) we obtain @xmath405 now , it is easy to find an arrangement for the vectors @xmath406 such that the chsh inequality is violated . the importance of these type of inequalities , of which bell s inequality was the first one , can not be overemphasized , as they have allowed for definitive experimental tests of quantum mechanics . the chsh inequality is also relevant in the context of hidden - variable models , and related no - go theorems  @xcite .    in the next section we will consider the problem of entanglement for systems of identical particles , for which the tools developed in the last sections will prove very useful . when combined , two of the most intriguing features of quantum theory -the intrinsic indistinguishability of identical particles and quantum entanglement- lead to formidable conceptual issues that have been addressed for years , and for which no generally accepted framework exists . the main source of problems comes from the fact that entanglement and related concepts have been studied mainly in cases where a subsystem decomposition coincides with a tensor product decomposition of the hilbert space . but precisely in the case of identical particles , the hilbert space is the antisymetric / symmetric subspace obtained from the action of the permutation group on a tensor product space . so precisely in this case decomposition into subsystems does not correspond to tensor product decomposition . this makes the use of partial trace a doubtful operation , leading to a clash with standard notions of entanglement . in  @xcite , we have shown how , using the representation theory of operator algebras , it is possible to obtain a generalized notion of entanglement . the universality of the approach , which can be applied to particles obeying any kind of statistics including bosons , fermions , parafermions and particles obeying braid statistics , provides a unifying view of entanglement , with many potential applications . the key idea is that , since subsystems can always be described in terms of subalgebras , the use of partial trace will be superseded by the more general notion of restriction of a quantum state @xmath211 to a subalgebra . then one can make use of the gns construction in order to find a hilbert space representation @xmath303 , as above . [ ex : entropy - irreducibility ] prove that the gns representation @xmath303 is irreducible precisely when the state @xmath211 is pure .    from this exercise , it follows that the condition for irreducibility of the representation @xmath303 is precisely that the von neumman entropy of the state @xmath211 vanishes and so , given a subsystem described in terms of a subalgebra @xmath407 , one may consider the restriction of the state @xmath211 to @xmath408 . this is a generalization of the notion of partial trace . hence , a generalized notion of entanglement emerges , based on the von neumann entropy of the restricted state . the properties of this entanglement measure are very closely tied to the gns - representation of the subalgebra and the restricted state . in other words , one finds that entanglement depends _ both _ on the state and the subsystem of the full system . this formalism , then , lends itself to the study of problems where partial trace loses its meaning , as is the case with systems of identical particles . let us consider a few illustrative examples . [ example:2x2matrix ] consider the algebra @xmath409 of @xmath410 matrices with complex entries , and let @xmath411 be a state on @xmath158 . of course , this can only happen if @xmath102 . let us then see how the gns representation is constructed , and how we can associate an entropy to the state @xmath412 . for this purpose , let us consider the matrix units @xmath413 @xmath414 . they generate the algebra , and provide a basis for the underlying vector space @xmath289 . they also fulfill the relations @xmath415 let us now check whether there will be null vectors . writing a general element @xmath213 as @xmath416 we compute @xmath417 from this expression we recognize that , if @xmath418 , then @xmath419 , whereas for @xmath420 or @xmath421 we obtain @xmath422 . in fact , if @xmath229 is different from zero or one , we must have both @xmath423 and @xmath424 . it follows that @xmath425 . from @xmath426\rangle = |[ab]\rangle\ ] ] and ( [ eq : e_ij ] ) we see that the representation is reducible , with 2 invariant subspaces generated , respectively , by @xmath427\rangle,|[e_{21}]\rangle\rbrace$ ] and @xmath428\rangle,|[e_{22}]\rangle\rbrace$ ] . from exercise [ ex : entropy - irreducibility ] we conclude that , in this case , the state has to be a mixed state . the cases @xmath420 and @xmath429 lead to a non - trivial null space , and from ( [ eq : null - states-2x2 ] ) we conclude that the dimension of these irreducibles is , in both cases , equal to 2 . it is left to the reader to obtain a pure state on the algebra @xmath430 such that @xmath412 is the restriction to @xmath431 . the next question we need to address is that of how to compute the entropy of the reduced state . since our algebras will in general be unital , there is a simple way to acomplish this , namely , given a ( unital ) @xmath0-algebra @xmath158 and a state @xmath211 on it , it is always possible to find a _ density matrix _ @xmath432 acting on the gns space @xmath295 , and such that @xmath433 for all @xmath215 in @xmath158 . from the decomposition of the gns space into irreducibles , @xmath434 we obtain _ projectors _ @xmath435 , with @xmath436 . from the definition of the inner product in @xmath295 we have @xmath437 | \pi_\omega ( a ) | [ \mathds 1_{\mathcal a}]\rangle$ ] . but then , using an orthonormal basis @xmath438 on @xmath439 , we can write : @xmath440 | \pi_\omega ( a ) |   [ \mathds 1_{\mathcal a}]\rangle\nonumber\\ & = & \langle [ \mathds 1_{\mathcal a } ] | \sum_k p^{(k)}\pi_\omega ( a ) |   [ \mathds 1_{\mathcal a}]\rangle\nonumber \\ & = & \langle [ \mathds 1_{\mathcal a } ] | \sum_k p^{(k)}\pi_\omega ( a)p^{(k ) } |   [ \mathds 1_{\mathcal a}]\rangle\nonumber\\ & = & \langle [ \mathds 1_{\mathcal a } ] | \sum_k p^{(k)}\pi_\omega ( a )   \sum_n |n\rangle \langle n| p^{(k ) } |   [ \mathds 1_{\mathcal a}]\rangle\nonumber\\ & = & \tr_{\mathcal h_\omega } ( \rho_\omega \pi_\omega(a)),\nonumber\end{aligned}\ ] ] with @xmath432 given by @xmath441 \rangle \langle [ \mathds 1_{\mathcal a}]| p^{(k)}.\ ] ] consider now a composite , bipartite system for which the hilbert space is of the form @xmath442 . if @xmath334 is a state on the algebra of bounded operators on @xmath443 that is obtained from a vector state @xmath444 , then its restriction to the subalgebra @xmath368 generated by elements of the form @xmath445 gives a state @xmath446 where @xmath447 is the _ reduced _ density matrix , defined through partial trace ( _ cf . _ exercise  [ ex : partial - trace ] ) , @xmath448 we therefore see that , for these kind of systems , partial trace equals restriction ( to some subalgebra ) . compute de von neumann entropy of the density matrix @xmath449 corresponding to the gns space of example [ example:2x2matrix ] and provide a physical interpretation . the real usefulness of the algebraic approach becomes apparent only when we consider situations where the hilbert space does not have a simple tensor product structure . this is what happens , _ e.g. _ , with systems of identical particles . because of the symmetrization postulate , the hilbert space contains only the symmetric ( for bosons ) or the antisymmetric ( for fermions ) subspaces of the many - particle hilbert space . as a consequence , one finds that states that from a physical point of view should not _ a priori _ be considered to be entangled , will have reduced density matrices with non - vanishing entanglement entropy  @xcite . in approaches like those of  @xcite or  @xcite , the description of subsystems is given by specifying suitable subalgebras . then , the restriction of a given state to the subalgebra provides a physically sensible generalization of the notion of partial trace . applying the gns construction to the restricted state , it is possible to study the entropy emerging from restriction and use it as a generalized measure of entanglement . let us briefly discuss how this entanglement measure can be computed in concrete cases . for this purpose , consider a hilbert space @xmath450 , assumed to correspond to the space of 1-particle states of a fermionic system . the full hilbert space is then the antisymmetric fock space @xmath451 obtained from @xmath452 . it decomposes as a direct sum of spaces of fixed number of particles . the @xmath453-particle hilbert space @xmath454 is just the antisymmetrized @xmath453-fold tensor product of @xmath455 . this is a kind of `` toy model '' for a fermionic quantum field theory , but many important features of a quantum field theory can be seen to appear already at this level . one of these features is the connection to the representation theory of clifford algebras , as explained in full detail in  @xcite . as is well - known , the clifford algebra of @xmath452 acts naturally on the exterior algebra @xmath456 which , in turn , is related to the fock space construction in the following way . let @xmath457 denote an orthonormal basis for @xmath452 , and denote with @xmath458 the corresponding annihilation ( creation ) operators ( _ cf . _  exercise [ ex : car - algebra - finite - dim ] ) . then , there is a vector space isomorphism between @xmath451 and @xmath459 , furnished by the correspondence @xmath460 this correspondence is behind the famous quote by e. nelson : _ `` first quantization is a mystery , but second quantization is a functor''_. in fact , given a self - adjoint operator @xmath23 on @xmath455 ( that is , a 1-particle observable ) , we obtain ( by functoriality ) an operator @xmath461 acting on fock space , whose restriction to the @xmath453-particle sector @xmath454 is given by @xmath462 taking the correspondence ( [ eq : nelson - correspondence ] ) into account , we obtain the following expression for @xmath461 in terms of creation / annihilation operators , @xmath463 where @xmath464 . in the physics literature , the operator @xmath465 is referred to as the _ second quantization _ of @xmath23 .    in the present quantum - mechanical context , where the number of particles is kept fixed , we want to focus our attention on the operator @xmath466 . one of the properties of this operator is that it preserves the symmetries of @xmath454 . furthermore , the map @xmath467 allows us to study subalgebras of 1-particle observables . the simplest example we can consider in order to illustrate entanglement issues for systems of identical particles is that of just two fermions . let us then consider , as done in  @xcite and in  @xcite , a 2-fermion system , where each fermion can be in a linear superposition of 4 basic states which , for the sake of physical interpretation , will be divided into internal and external degrees of freedom . so we describe 1-particle states in terms of a set of ( fermionic ) creation / annihilation operators @xmath468 , where @xmath215 stands for _ `` left '' _ , @xmath469 for _ `` right '' _ ( the external degrees of freedom ) and @xmath470 for spin up and down ( the internal degrees of freedom ) . hence , in this case we have @xmath471 and , for the 2-fermion space , @xmath472 . an orthonormal basis for @xmath473 is given by the vectors @xmath474 the two - particle algebra @xmath158 of observables is thus isomorphic to the matrix algebra @xmath475 .    for @xmath476 , the corresponding state @xmath477 is given by latexmath:[$\omega_{\theta } ( \alpha)= \langle \psi_{\theta }    we choose the subalgebra @xmath408 to be the one generated by @xmath479 , @xmath480 , @xmath481 and @xmath482 . physically , it corresponds to the subalgebra of one - particle observables corresponding to measurements at the left location .    consider the restriction of @xmath477 to @xmath408 and study the gns representation corresponding to this choice . for @xmath483 you should obtain vanishing entropy , in contrast to the result @xmath484 obtained via partial trace for states with slater rank one  @xcite . the hamiltonian for the ( quantum ) ising chain in a transverse field is given by @xmath485 an important aspect when solving the model and studying its solution is the type of boundary conditions considered . we will be interested in both open as well as periodic boundary conditions . the first step in the solution of this model is the so - called wigner - jordan transformation , that allows us to express all spin operators in terms of _ fermionic _ creation / annhilation operators :    as can be easily checked , the operators defined by @xmath486 where @xmath487 , are fermionic operators .    [ ex:1.1 ] check that the operators defined above indeed satisfy the car algebra : @xmath488 , @xmath489 . the inverse transformation is given by @xmath490    [ ex:1.2 ] check that ( [ eq : wigner - jordan - inverse ] ) is the inverse transformation to ( [ eq : wigner - jordan - def ] ) .    with this we obtain , for the interaction terms of the hamiltonian , @xmath491 and so @xmath492 the takes the form @xmath493 the term containing the external field @xmath229 is diagonal in this basis . the constant term @xmath494 coming from the last sum is usually disregarded , because its only effect is to shift the energy spectrum . we will nevertheless keep all terms , in order to be able to compare with numerical solutions in the spin basis , for small values of @xmath495 .    expanding all terms in ( [ eq : h - ising2 ] ) and moving all creation operators to the left , we obtain ( _ cf . ( 10.14 ) in  @xcite ) : @xmath496 the point of using the wigner - jordan transformation for this model is that the hamiltonian becomes `` almost '' diagonal . by this we mean that it is a sum of local , _ quadratic _ expressions in the creation and annihilation operators . as we will see , such models can be exactly solved . we now write @xmath492 in a suggestive matrix notation . arranging all creation and annihilation operators in rows and columns , we can write @xmath492 as a kind of quadratic form . for example , we have , for @xmath497 , @xmath498 the expression corresponding to the same term for @xmath499 is then @xmath500 for arbitrary @xmath495 , we may define the following @xmath501 matrices : @xmath502 the matrix elements of these two matrices can be written as follows : @xmath503 hence , the hamiltonian takes the following form : @xmath504 -\lambda n.\ ] ]      for several reasons , including the study of edge states , it is instructive to explore the explicit solution of this model for _ open _ boundary conditions . we start with the ising hamiltonian written in the form ( [ eq : h - ising3 ] ) . any model that can be written as a quadratic form can be expressed in this way , the `` only '' difference being the explicit form of the matrices @xmath23 and @xmath3 ( notice that , in order for @xmath492 to be hermitian , @xmath23 has to be symmetric and @xmath3 antisymmetric ) . in principle , thus , the method presented below ( following the work of lieb , schultz and mattis  @xcite ) can be applied to any such model .    for periodic boundary conditions , it is usually more convenient to take into account translation invariance and hence to introduce fourier transformed operators . but for open boundary conditions , translation invariance is `` broken '' and then it is a good idea to start right away with a bogoliubov transformation , as explained below . recall that the operators @xmath505 defined in ( [ eq : wigner - jordan - def ] ) obey fermionic canonical anti - commutation ( car ) relations ( _ cf . _  exercise  [ ex:1.1 ] ): @xmath506 in general , a bogoliubov transformation is a mapping induced by a change of basis on the one - particle hilbert space ( unitary transformation ) , its main effect being to provide a new set of creation / annihilation operators for which the hamiltonian ( [ eq : h - ising3 ] ) becomes diagonal . consider , then , a new set of operators given by @xmath507 where @xmath508 and @xmath509 are @xmath501 matrices to be chosen so that    * the new operators satisfy the same car algebra : @xmath510 * the hamiltonian becomes _ diagonal _ in the new basis : @xmath511 ( with @xmath257 some constant ) . [ ex:1.3 ] show that the requirement ( [ eq : bogol - car ] ) leads to the following conditions : @xmath512    [ ex:1.4 ] compute the trace of @xmath492 in two different ways , in order to show that the constant term in ( [ eq : h - ising - diagonal ] ) is given by @xmath513    the second condition above , eq . ( [ eq : h - ising - diagonal ] ) , will lead to an eigenvalue problem for @xmath508 and @xmath509 , the solution of which amounts -in principle- to the solution of the full problem . now we compute the commutator @xmath514 $ ] in two different ways , once using ( [ eq : h - ising3 ] ) and once using ( [ eq : h - ising - diagonal ] ) . this leads to the following set of equations : @xmath515 in order to solve this eigenvalue problem , it proves convenient to introduce new matrices @xmath516 and @xmath517 , as follows : @xmath518 if we now define for each @xmath453 a vector @xmath519 , the @xmath520 component of which is given by @xmath521 , and similarly for @xmath517 , we find that ( [ eq : ga - hb ] ) can be written as follows : @xmath522 or , equivalently , as @xmath523 with this we have reduced our problem from the diagonalization of a @xmath524 matrix to that of diagonalizing two @xmath501 ones . as mentioned before , this would be a very easy task if we would have chosen periodic boundary conditions . the reason being that for periodic boundary conditions the matrices @xmath525 are _ toeplitz_. but for open boundary conditions the matrices @xmath23 and @xmath3 are given by ( [ eq : ab - open ] ) , so that @xmath526 and @xmath527 in order to solve ( [ eq:(a - b)(a+b ) ] ) , we propose the following ansatz : @xmath528 with @xmath529 and @xmath530 constants to be determined . the eigenvalue equation will then give 3 independent equations . the first one is obtained by equating the @xmath531 component of @xmath532 with @xmath533 , for @xmath534 . all these choices of @xmath5 yield the same equation ( by enforcing the vanishing of the coefficients of @xmath529 and @xmath530 ) , namely , @xmath535 this is ( almost ) the spectrum of our problem . we still need to find what are the allowed values of the label `` @xmath453 '' . this is done by considering the two other cases ( @xmath536 and @xmath537 ) , that give a system of equations for @xmath529 and @xmath530 : @xmath538 the non - trivial solution @xmath539 is obtained provided the determinant of this matrix vanishes . this condition is equivalent to @xmath453 being solution of the following transcendental equation , for which @xmath540 has to be assumed : @xmath541 notice that , since we must have @xmath495 eigenvectors , we expect this equation to have @xmath495 roots . the behavior of these roots as a function of @xmath229 is quite relevant ; comparison with the periodic chain allows for the recognition of edge states .      in the case of periodic boundary conditions , we extend the sums in ( [ eq : h - ising - transv ] ) to @xmath542 , adopting the convention that @xmath543 ( @xmath544 ) . this generates a boundary term that couples the first spin operator to the last one : @xmath545 we already know how to write the first two terms of this hamiltonian in terms of creation and annihilation operators . let us therefore consider the last term : @xmath546 . here it is convenient to consider the _ parity _ operator , @xmath547 , where @xmath548 is the _ number _ operator . using the identities @xmath549 and ( [ eq : wigner - jordan - inverse ] ) we obtain : @xmath550 with this we can write @xmath492 in terms of fermionic operators : @xmath551 an important fact ( that can be easily checked ) is that the parity operator commutes with @xmath492 : @xmath552 = 0 . \nonumber\ ] ] thus , it is possible to diagonalize the hamiltonian separately in sectors of even and odd numbers of `` particles '' . the eigenvalues of @xmath547 are of the form @xmath553 , with the plus sign for states with an _ even _ number of particles and the minus sign for states with an _ odd _ number of particles . we can , as in the previous case , write @xmath492 as a quadratic form in the fermion operators ( _ cf . _ ( [ eq : h - ising3 ] ) ) , the only difference being the explicit form of the matrices @xmath23 and @xmath3 , for which we now get : @xmath554 in contrast to ( [ eq:(a - b)(a+b ) ] ) or ( [ eq:(a+b)(a - b)matrix ] ) , the matrix @xmath555 is now toeplitz : @xmath556 the eigenvalue problem ( [ eq:(a - b)(a+b ) ] ) can again be solved using the ansatz ( [ eq : psi_kl ] ) . the eigenvalues are again given by ( [ eq : lambda_k ] ) , with the difference that the allowed values of @xmath453 can now be explicitly given ) . ] , as follows from the following exercise .    [ ex:1.5 ] show that , in the even parity sector ( @xmath557 ) , the allowed values of @xmath453 are given by @xmath558 whereas in the odd parity sector ( @xmath559 ) they are given by @xmath560    [ ex:1.6 ] explain how the @xmath453 s can be made to take positive and negative values and how then we obtain an explicit solution for the matrix @xmath516 , of the form @xmath561    it should by now be clear that the solution of the eigenvalue problem is much easier in the periodic case . we have chosen to discuss the open chain mainly because of the role that boundary effects play in connection to symmetry breaking and also for studies of surface effects  @xcite . now we are going to take advantage of the translational symmetry in the periodic case . as discussed below , if instead of applying a bogoliubov transformation right after transforming the spin operators to fermion operators via the jordan - wigner transformation , we perform a fourier transformation as an intermediate step , we quickly obtain a much more elegant solution , that can be easily generalized to treat the limiting case of an infinite chain . here we will again make use of the fact that the hamiltonian commutes with the parity operator . looking back at ( [ eq : h - with - bdry - term ] ) , we realize that it is possible to express @xmath492 in a more compact form if we introduce boundary conditions of the fermion operators , depending on the parity sector we are interested in . in fact , notice that if we choose @xmath562 , then the boundary term can be included in the first sum , just by setting @xmath563 . but if we want to do the same in the sector @xmath564 , we will need @xmath565 . in view of our previous discussion , it is clear that the choice @xmath562 is more convenient computationally ( the matrices @xmath525 are circulant in that case ) . but if we are interested in the _ ground state _ , then we must consider the other choice ( @xmath566 ) . this can be seen if we compare the ground state energies of the fermionic hamiltonian in each sector . as can be easily checked , the lowest energy eigenvalue indeed comes from the sector @xmath566 . the difference between them decreases with increasing @xmath495 and eventually disappears in the thermodynamic limit . it is for this reason that the boundary term is usually disregarded . we nevertheless feel that the price to pay if we keep all terms is actually very low , and in return we can have complete control over the spectrum and its degeneracies  @xcite . [ ex:1.7 ] making use of the solution of the eigenvalue problem for both periodic and antiperiodic boundary conditions , plot the lowest energy eigenvalues for @xmath567 as functions of @xmath229 and @xmath495 . we therefore define @xmath568 in this way the hamiltonian becomes @xmath569 notice that the sum now goes over @xmath570 . so we see that we can describe _ both _ sectors using the _ same _ quadratic form , the choice of parity being now encoded in the boundary conditions for the fermion operators . it should be remarked that when we consider anti - periodic boundary conditions for the fermions we are still considering periodic boundary conditions for the _ spin chain_. anti - periodicity for the fermions in this context is only related to the choice of a negative eigenvalue for the parity operator . let us now introduce the fourier transformed operators @xmath571 the choice of the phases @xmath572 must be made in such a way that the car algebra is preserved , but in addition it has to imply the boundary condition @xmath573 . therefore , the phases @xmath572 will also depend on the parity sector . [ ex:1.8 ] show that in the _ even _ parity sector ( @xmath574 ) , the phases are given by @xmath575 whereas in the _ odd _ parity sector ( @xmath576 ) they are given by @xmath577    a word of caution is perhaps in order : in principle , @xmath453 is an integer that takes values in the range @xmath578 . nevertheless , due to the translational symmetry , @xmath453 actually belongs to a `` brillouin zone '' , meaning with this that there is no difference between @xmath453 and @xmath579 . so , for instance , for @xmath566 we have that @xmath580 whenever @xmath581 is an integer or , in other words , when @xmath582 for this reason , even if the allowed values of @xmath453 are originally in the range @xmath583 , we will allow @xmath453 to take any integer value , provided we interpret this ( for @xmath566 ) in the sense of ( [ eq : mod - n ] ) . with this convention we then obtain , for example : @xmath584 a similar remark applies to the case @xmath562 . for the remaining part of this section we will confine ourselves to the even parity sector and so @xmath566 will be tacitly assumed . we can now go back to the hamiltonian ( [ eq : h - fermions - periodic ] ) and insert there the new operators defined by ( [ eq : fourier - dk ] ) . as a result , we obtain the following form of the hamiltonian : @xmath585.\ ] ] this form of the hamiltonian is indeed very convenient as it suggests that a diagonalization by means of a bogoliubov transformation that only mixes @xmath586 with @xmath587 should be possible . for this reason we propose the following transformation : @xmath588 with @xmath589 ( possibly complex ) coefficients to be found . imposing the condition @xmath590 we readily obtain @xmath591 . furthermore , it is easy to check that the choices @xmath592 enforce @xmath593 . inverting ( [ eq : c_k ] ) and inserting the result in ( [ eq : h - en - terminos - de - d_k ] ) , we obtain : @xmath594 + \mbox{h.c . } \right ) {   } }    \nonumber\\ & & { } + 2 \sum_{k=1}^n c_k^\dagger c_k \big [ ( \lambda -\cos \phi_k ) (   |\alpha_k|^2-|\beta_k|^2 ) + i \sin\phi_k ( \bar{\alpha}_k \beta_k   -    \alpha_k \bar{\beta}_k ) \big ] { } \;\;\;\nonumber\\ & &   { } + \sum_{k=1}^n\big (    2(\lambda -\cos\phi_k ) |\beta_k|^2   + i \sin \phi_k ( \alpha_k   \bar{\beta}_k- \bar{\alpha}_k\beta_k ) -\lambda \big ) . { } \end{aligned}\ ] ] vanishing of the coefficients in the first sum leads to @xmath595 recalling that @xmath591 , we see that the right hand side of this expression simplifies if we parameterize @xmath596 and @xmath597 as follows : @xmath598 in fact , with this choice ( [ eq : rl1 ] ) simplifies to @xmath599 we can now introduce a normalization factor @xmath600 we will multiply @xmath492 by a factor 1/2 , in order to simplify some expressions . ] by means of @xmath601 this leads to @xmath602 with these definitions , the remaining terms in ( [ eq : rl2 ] ) simplify further yielding the desired diagonal form for the hamiltonian : @xmath603    [ ex:1.9 ] verify all the computations leading from ( [ eq : h - fermions - periodic ] ) to ( [ eq : rl3 ] ) .    * the ground state . * having brought the hamiltonian to the diagonal form ( [ eq : rl3 ] ) , we now proceed to find an expression for the ground state . since @xmath600 has been chosen to be positive ( or zero ) for all values of @xmath453 , the ground state @xmath604 will be given by the condition @xmath605 . let @xmath606 be the `` vacuum '' state for the operators @xmath607 , defined by the condition @xmath608 . observe now that if we define @xmath609 then we get @xmath610 for all @xmath453 . since @xmath611=0 $ ] , one would think that @xmath612 does the job , but we must take into account that @xmath613 whenever @xmath614 . if @xmath495 is even , this means that _ each _ @xmath615 appears _ twice _ in @xmath616 and then an easy calculation shows that @xmath617 does not vanish . hence , we must be careful to include in the product only one instance of each @xmath615 .    to do this , we distinguish two cases : @xmath495 even and @xmath495 odd . restricting the domain of @xmath572 to the interval @xmath618 we get , for @xmath495 even , @xmath619 for @xmath620 ( also for @xmath621 ) and @xmath622 for @xmath623 . now , for each @xmath453 such that @xmath619 , there is _ exactly _ one @xmath624 with @xmath625 and such that @xmath626 . in fact , let @xmath453 be such that @xmath619 and put @xmath627 . it follows that @xmath625 and @xmath628 , so that @xmath626 . we therefore see that , _ for even _ @xmath495 , the ground state is given by @xmath629 writing @xmath630 , we may as well consider @xmath453 to be such that @xmath631 . the ground state , then , takes the form @xmath632    for the case @xmath495 odd , write @xmath633 . in this case all the @xmath615 appear twice in @xmath616 , _ except _ when @xmath634 . when @xmath634 we have @xmath635 . this , together with the fact that @xmath636 , implies that @xmath637 , so we must exclude it from the product . in analogy to the previous case , for each @xmath453 for which @xmath638 we can find exactly one @xmath624 such that @xmath639 and @xmath626 , namely @xmath640 . now , for @xmath641 , we have @xmath642 , whereas for @xmath643 , we have @xmath644 . the result is that , _ for odd _ @xmath495 , the ground state is given by @xmath645 the operator @xmath646 has to be included in ( [ eq : gs - ising - odd ] ) in order to ensure that @xmath647 ( recall that in this case @xmath648 ) . having obtained the exact spectrum of the model , as well as the ground state both for open and periodic boundary conditions , it is now possible -in principle- to obtain all relevant correlation functions and relate them to physical observables like magnetization , susceptibilities , and so on . as is well - known , the quantum ising chain does not present any phase transition at finite temperature , but it certainly is one of the paradigmatic examples of a system displaying a quantum critical point . in fact , in the thermodynamic limit there is a quantum phase transition occurring when the external field @xmath229 approaches the critical value @xmath649 . in this notes we will review an approach to the study of this quantum phase transition from the point of view of _ entanglement_. we will also be interested in certain geometric and topological properties of the ground state that are relevant for this quantum phase transition . let us start by considering the model ( [ eq : h - ising - transv ] ) for an open , two - site chain ( @xmath497 ) . putting @xmath650 , we find that the ground state is given in the spin basis by @xmath651 taking the partial trace with respect to one of the sites , we obtain @xmath652 which in terms of the basis @xmath653 takes the following simple matrix form : @xmath654 figure [ fig:1.1 ] displays a plot of the eigenvalues of @xmath4 as a function of @xmath229 .    ]    as we shall see , the behavior of this _ entanglement spectrum _ as the size of the chain increases will provide important information regarding the quantum phase transition in this model . [ ex:1.10 ] follow the steps outlined in section [ sec : openboundaryconditions ] to show that for @xmath497 the matrices @xmath508 and @xmath509 introduced in ( [ eq : g - h - defined ] ) are given by@xmath655 use this in order to show that the condition @xmath656 leads exactly to ( [ eq : gs - ising - two - sites ] ) . as we have seen , in this simple case ( @xmath497 ) the density matrix for the reduced state can be easily computed . in spite of the fact that the model can be exactly solved , the computation of correlation functions ( as those involved in the computation of the reduced density matrix ) for arbitrary values of @xmath495 is a non - trivial task . we will therefore present , following  @xcite , a method that allows us to obtain the spectrum of @xmath4 for arbitrary values of @xmath495 and that can be used for any quadratic hamiltonian . as a preparation for the general case , we first explain the idea using our very simple example of a two - site chain , where all calculations can be explicitly carried out . let @xmath158 be the car algebra generated by the two fermionic operators @xmath657 and the unity @xmath184 . we consider the `` fock representation '' where @xmath658 acts on the state @xmath659 . this state is cyclic and therefore we can write the ground state ( [ eq : gs - ising - two - sites ] ) in terms of it ( _ cf . _  exercise [ ex:1.10 ] ) : @xmath660 as discussed in section [ sec : algebras - and - states ] , we can use it to define a state @xmath412 in the _ algebraic sense _ : @xmath661 this is , of course , a pure state . nevertheless , if we _ restrict _ it to the subalgebra @xmath368 that is generated by @xmath184 and @xmath662 ( `` a half - chain '' ) , then we find that in general the resulting state @xmath663 we expect that for any @xmath664 @xmath665 will hold , with @xmath666 @xmath667 , both traces being taken on the hilbert space corresponding to the half - chain ( which in this case has a basis given by @xmath668 ) . finding @xmath669 amounts to finding the spectrum of @xmath4 and this , in turn , can be used to compute the entanglement entropy . the specific form of @xmath4 as the exponential of a quadratic form is to be expected in view of the fact that the ground state is gaussian . in order to find @xmath669 , we notice that it is possible to compute the left hand side of ( [ eq : a1-n=2 ] ) . the only relevant case is @xmath670 , so let us define @xmath671 . using ( [ eq : gs - n=2 ] ) , we get : @xmath672 where we still keep the notation @xmath673 . on the other hand @xmath674 it follows that @xmath675 or , equivalently , @xmath676 since @xmath4 is already diagonal in the basis @xmath668 , we obtain : @xmath677 which coincides with ( [ eq : rho - n=2 ] ) . notice , in passing , that @xmath678 can also be obtained directly from @xmath509 . in fact , a small computation shows that @xmath679 . this last remark gives a hint towards the solution for the general case : as we shall see , the spectrum of @xmath4 in the general case can be extracted from an eigenvalue problem that uses the matrices @xmath509 and @xmath508 as input . let us , then , start by considering the algebra @xmath158 generated by @xmath184 and by all operators @xmath680 , with @xmath681 . again , the ground state @xmath604 gives rise to a state @xmath682 . we are interested in the restriction of this state to the subalgebra @xmath683 generated by @xmath684 . by the same argument as before , we expect this new state @xmath685 to be of the form @xmath686 ( @xmath687 ) , with @xmath688 and @xmath689 for the ising model , the matrices @xmath690 and @xmath167 ( yet to be determined ) will be real , so that @xmath690 has to be symmetric , and @xmath167 antisymmetric . now we define  @xcite , for @xmath691 the correlation functions @xmath692 using ( [ eq : g - h - defined ] ) one shows that @xmath693 and @xmath694 . let us first consider the simpler case @xmath695 . in this case @xmath696 can be diagonalized by a simple transformation of the form @xmath697 where the matrix @xmath698 is such that @xmath699 . solving the eigenvalue problem @xmath700 and introducing the convention @xmath701 ( @xmath702 in the @xmath520 entry and @xmath703 everywhere else ) , we set @xmath704 , _ i.e. _ , the rows of @xmath698 are the eigenvectors @xmath705 . we then obtain @xmath706 it follows that @xmath707 , so that @xmath708 . using ( [ eq : b - gamma - a ] ) we then obtain @xmath709 , which is equivalent to the matrix identity @xmath710 . this can also be written in the following form : @xmath711 if @xmath712 , we replace ( @xmath713 ) by @xmath714 where now we require ( _ cf . _  exercise  [ ex:1.3 ] ) : @xmath715 once again , the matrices @xmath698 and @xmath716 should be such that @xmath4 becomes diagonal , _ i.e. _ , @xmath717 should hold . the eigenvalue problem to be solved is @xmath718 using these eigenvectors as rows for the matrices @xmath719 and @xmath12 and putting @xmath720 we obtain the desired diagonal form for @xmath696 . as in the previous case , we can now write all correlation functions that involve the operators @xmath680 in terms of correlation functions that only involve the operators @xmath721 .    [ ex:1.11 ] obtain the following two relations : @xmath722 and use them in order to prove the following identity : @xmath723 where @xmath65 and @xmath724 are the @xmath725 matrices with components given by ( [ eq : c_ij - f_ij ] ) .    from this exercise we obtain , making use of ( [ eq:(k+m)(k - m ) ] ) , @xmath726     sites as a function of the external field @xmath229 ]    we will now make use of this expression in order to obtain the entanglement spectrum and the entanglement entropy of the ground state of the ising chain when @xmath728 , for various values of @xmath495 and @xmath229 . in fig . [ fig:1.2 ] we plot the entanglement spectrum ( in logarithmic scale ) for a chain of @xmath729 sites , when the ground state is restricted to a half - chain . we notice that , in contrast to the case @xmath497 , the gap between the first two eigenvalues is now closed for almost all values of @xmath229 below @xmath649 .    in fig . [ fig : subfigures ] we compare the entanglement spectrum for an ising chain of @xmath729 sites against the spectrum of a chain of @xmath730 sites . it is apparent that for @xmath730 the gap near @xmath731 has almost closed .     and @xmath730 , showing how the gap at @xmath421 closes as the number of sites increases.,scaledwidth=150.0% ]    this gap , when evaluated at the critical value @xmath731 , bears the name of _ schmidt gap_. we will use the notation @xmath732 for the schmidt gap . de chiara and collaborators have shown , using finite - size scaling , that the schmidt gap can be interpreted as an _ order parameter _  @xcite . in fact , from a finite size scaling analysis , one can in fact obtain numbers @xmath733 and @xmath734 such that when we plot @xmath735 versus @xmath736 , all points collapse to a single curve , irrespective of the value of @xmath495 chosen . this result was obtained in  @xcite using a dmrg algorithm . figure [ fig : catalina ] , obtained by c. rivera using the same dmrg technique  @xcite depicts the resulting curve , for which points corresponding to different lengths of the chain ( ranging from @xmath737 to @xmath738 ) are seen to collapse to a single curve . the values of @xmath733 and @xmath734 for which this is achieved give values for the critical exponents which are very close to the actual values ( @xmath739 and @xmath740 ) .        the entanglement spectrum can also be used to compute the entanglement entropy and its dependence on the size of the subsystem chosen . a numerical evaluation of this entropy gives the result expected from conformal field theory , that is , at the critical point , the entanglement entropy follows a logarithmic behavior , of the form @xmath741 where @xmath742 denotes the central charge of the corresponding cft . in this case ( ising chain ) , numerical evaluation of the entropy for different chain sizes gives a value of @xmath743 , in accordance with the predictions of cft . the xy model is a generalization of the ising model . the model also represents a spin chain with nearest - neighbor interactions . its hamiltonian is given by @xmath744 there are two parameters in this model . as in the ising case , @xmath229 represents an external field . the parameter @xmath698 is an anisotropy parameter . notice that if @xmath745 , we recover , up to an overall factor of @xmath746 , the ising hamiltonian ( [ eq : h - ising - transv ] ) . our interest in this model lies in the fact that according to the values of the parameters @xmath698 and @xmath229 , we may study critical regions corresponding to _ different _ universality classes : if @xmath747 , there is a critical line determined by @xmath748 that corresponds to the ising universality class , with central charge @xmath743 . on the other hand , for @xmath749 the whole line @xmath750 corresponds to the @xmath751 universality class , with central charge @xmath752 . the solution of this model can be obtained in the same fashion as we did in the case of the ising model . the first step , then , consists in using the wigner - jordan transformation to bring the hamiltonian to a quadratic form in fermionic operators . the result one obtains is @xmath753 with @xmath754 , the fermionic number operator . since we are interested in the ground state , we consider the even parity sector and proceed , as with the ising model , with a fourier transformation , followed by a bogoliubov transformation . for odd @xmath495 ( @xmath755 ) and with @xmath756 we obtain : @xmath757 we can now consider a bogoliubov transformation , as in ( [ eq : c_k ] ) , @xmath758 and obtain the following diagonal form for the hamiltonian : @xmath759 with @xmath760 defining @xmath761 we obtain the following explicit form for the ground state @xmath762 : @xmath763 where @xmath606 is the vacuum state with respect to the operators @xmath607 . the xy model has a rich phase diagram , and this provides an opportunity to check the behavior of entanglement entropy at the critical point and its relation to the central charge . in the previous section we focused on a finite - size scaling analysis for the ising model , which in turn brought up the difficulty of having to compute correlation functions for finite size chains . in this section , we will only be interested in properties in the thermodynamic limit , _ i.e. _ , we will discuss the behavior of the entanglement entropy of a sub - chain of length @xmath764 , considered as a subsystem of a chain of infinite length , or the geometric phase of the ground state associated to closed loops around the critical point . calculations in the thermodynamic limit are greatly simplified because discrete sums become integrals that can be readily evaluated . let us now very briefly describe how to obtain the entanglement entropy of a sub - chain of size @xmath764 in the thermodynamic limit . we will follow the approach of vidal _ et al . _ @xcite , which amounts to the real ( _ e.g. _  majorana ) version of the method discussed in section [ sec : entanglement - properties - gs ] . we begin by replacing the fermion operators ( @xmath680 ) appearing in ( [ eq : h_xy ] ) by new , majorana fermion operators ( @xmath765 ) , defined as follows : @xmath766 these majorana operators satisfy commutation relations of the form @xmath767 . now , consider the correlation matrix @xmath768 , where the expectation value is taken with respect to the ground state . the matrix @xmath769 can be expressed as a block matrix , with the block @xmath770 being given by the @xmath410 matrix @xmath771 ( @xmath345 and @xmath5 range from @xmath703 to @xmath772 , @xmath495 being the size of the full chain ) , where @xmath773 in the thermodynamic limit ( @xmath774 ) the coefficient function @xmath775 takes the following compact form : @xmath776 the restriction of the correlation matrix to the sites belonging to a sub - chain of size @xmath764 is then a @xmath725 matrix @xmath777 which is obtained just by deleting the entries of @xmath769 that do not correspond to sites in the sub - chain .    following the method of section [ sec : entanglement - properties - gs ] as well as vidal et al . @xcite , show that the entanglement entropy corresponding to the restriction of the full chain to a sub - chain of @xmath764 sites is given by the formula @xmath778 where @xmath492 denotes the shannon entropy as defined in ( [ eq : h(lambda ) ] ) and @xmath779 are the positive eigenvalues of @xmath780 . figure [ fig : vidal ] displays the behavior of the entropy as a function of @xmath764 . one of the curves ( dashed ) shows the results for the xx chain , which corresponds to @xmath749 . from the logarithmic dependence of the entropy we extract the value @xmath752 for the central charge in this model . the continuous curve shows the entropy for the critical case @xmath781 , which belongs to the ising universality class . in this case , the value obtained for the central charge is @xmath743 . the dots correspond to the non - critical case @xmath782 . we can see that in this case the entropy saturates with @xmath764 . thus , the results confirm the expected behavior of the entropy as predicted by cft . is represents the size of the sub - chain . for model parameters corresponding to critical phases , we observe a logarithmic behavior for the entanglement entropy . outside the critical point , the entropy saturates . ] in section [ sec:2 ] we mentioned that the _ canonical commutation relations _ ( ccr ) lead to a @xmath0-algebra , whose representation theory plays a very important role in quantum field theory . let us recall that the operators @xmath177 and @xmath178 defined in ( [ eq : u(a)v(b)def ] ) satisfy the so - called weyl form of the ccr ( _ cf . _ exercise  [ ex : weyl - ccr ] ) . the first thing we want to do is to recognize that any implementation of the ccr in their weyl form corresponds in fact to the construction of a representation of a certain group ( the heisenberg group ) . the heisenberg group of @xmath783 , whose underlying set is @xmath784 , is the group defined by the following operation : @xmath785        the key point regarding these commutation relations is actually their relation to the ( canonical ) symplectic structure of the underlying classical phase space , @xmath790 . in fact , if for @xmath791 we define @xmath792 we obtain the following identity : @xmath793 where @xmath267 and @xmath794 denote elements of the _ symplectic vector space _ @xmath790 , and @xmath112 the standard symplectic form . notice that we have the following relations between all these operators : @xmath795    ( _ cf . _  @xcite ) let @xmath789 be a real vector space and @xmath796 a symplectic form ( _ i.e. _ a bilinear , skew - symmetric , non - degenerate form ) . a @xmath149-algebra @xmath797 is called a weyl @xmath149-algebra of @xmath798 if there is a family @xmath799 of `` generators '' such that      in  @xcite it is proved that every symplectic vector space @xmath798 determines uniquely a weyl @xmath149-algebra , up to @xmath149-isomorphism . also , if @xmath804 is a symplectomorphism between two symplectic vector spaces , then the corresponding weyl @xmath149-algebras are isomorphic . a most important fact is that @xmath797 can be completed to a @xmath0-algebra , the weyl @xmath0-algebra . it is a fundamental result , due to stone and von neumann , that when the dimension of @xmath789 is finite ( and hence necessarily even ) , there is essentially only one representation of the ccr , which can be taken to be the standard schrdinger representation  @xcite . but in infinite dimensions uniqueness is lost , and so _ inequivalent _ representations do exist . this fact is closely related to the non - uniqueness of a vacuum state for a free quantum field in a curved spacetime background . the following example illustrates how the construction of a ( free , bosonic ) quantum field runs in parallel to the previous discussion of ccr , the `` only '' ( ! ) difference here being that the symplectic vector space we are dealing with is infinite dimensional . let @xmath805 denote a globally hyperbolic spacetime , and consider the klein - gordon equation on that background : @xmath806 let @xmath807 denote a fixed cauchy surface , to which the initial data for the solutions of the klein - gordon equation will be referred and consider the following vector space , a _ space of solutions of the k - g equation _ : @xmath808 it is a most remarkable fact that the classical field equation comes equipped with a natural symplectic structure . in fact , if @xmath809 is any space - like ( cauchy ) hypersurface , and @xmath810 denotes the unit normal , then on the vector space @xmath811 defined above we can define the following symplectic form : @xmath812 for @xmath813 , let @xmath23 and @xmath814 denote the advanced and retarded fundamental solutions of the k - g equation . that is , for @xmath815 we have @xmath816 where the support of @xmath817 lies in the causal past of the support of @xmath59 ( and analogously for @xmath818 ) . then @xmath819 gives a map @xmath820 the ccr for the quantum field ( in its _ smeared form _ ) then take the form @xmath821=-i \sigma(ef , eg ) . \nonumber\ ] ] just as in non - relativistic quantum mechanics , the operators obeying ccr are unbounded , and so it is convenient to go to the weyl form , which leads us back to the weyl @xmath0-algebra , this time the one corresponding to the infinite dimensional vector space @xmath822 .      representation spaces for the weyl algebra of the previous example are bosonic fock spaces . for the classical dirac field , the transition to the corresponding quantum description can be performed in a completely analogous way , only that in this case , instead of an antisymmetric form , a symmetric one is used which , again , comes `` for free '' as part of the structure of the ( vector space ) of solutions of the classical field equation . it is an interesting fact that the existence of a natural bilinear antisymmetric form in the scalar field case ( and of a symmetric one in the case of the dirac field ) leads to quantum fields that obey the physically correct relation between spin and statistics . in very general terms , quantization of free fields can be formulated as follows . suppose we are given a real vector space @xmath789 together with a non degenerate ( anti-)symmetric bilinear form @xmath823 . if in addition we choose a complex structure @xmath824 which is compatible with @xmath825 ( meaning @xmath826 holds ) , we can complexify and obtain a complex vector space @xmath827 . the bilinear form @xmath825 can then be used to define an inner product on @xmath827 , making it a hilbert space . according to whether the bilinear form is symmetric or antisymmetric , one then considers the antisymmetric or the symmetric subspace of the tensor algebra of @xmath827 , call it @xmath828 . endomorphisms of @xmath827 may be lifted to @xmath829 in a natural ( _ i.e. _ functorial ) way . these liftings are then interpreted as the `` second quantization '' of observables ( self - adjoint operators in @xmath827 ) and symmetries ( unitary operators ) . + in this section we will focus on the fermionic case , following the beautiful treatment of the subject presented in  @xcite , which makes strong use of clifford algebras and spin group representations . in the next section we will explore the connection between second quantization ( as presented here ) and quantum criticality in spin chains ( as discussed previously ) . let us start by first considering the simple case where @xmath830 . if @xmath789 is a finite dimensional real vector space , and @xmath508 a positive definite symmetric bilinear form on it , we can construct the corresponding clifford algebra @xmath831 . as shown in  @xcite , it is possible to obtain a concrete realization of the complexified clifford algebra @xmath832 as a subalgebra of @xmath833 , but the corresponding representation is not irreducible . the situation changes if we assume @xmath789 has even dimension , say @xmath834 . then we may choose an orthogonal complex structure , that is , a linear operator @xmath835 such that @xmath836 and @xmath837 for all @xmath838 , and regard @xmath789 as a complex vector space @xmath827 of dimension @xmath839 . the scalar product on @xmath827 is given by @xmath840 it then turns out that the exterior algebra @xmath841 is an irreducible clifford module . we call this module _ fock space _ and use the notation @xmath842 for it . the action of @xmath832 on @xmath842 is obtained by defining  creation " and  annihilation " operators @xmath843 acting on @xmath842 as follows : @xmath844 from these definitions , we obtain the anti - commutation relations @xmath845 from which @xmath846 follows . hence , the map @xmath847 may be extended to @xmath848 by making use of the universal property of the clifford algebra . the scalar product @xmath849 on @xmath827 induces one on @xmath842 , which on basis elements is given by @xmath850 thus making @xmath842 a ( finite dimensional ) hilbert space . it may be checked that @xmath851 is really the adjoint of @xmath852 , with respect to this scalar product . our interest is to lift ( or quantize ) operators @xmath853 to operators acting on fock space . in the present context , this is done by means of maps @xmath854 defined as the graded operators obtained from @xmath855 and @xmath856        given any @xmath859 and @xmath860 any _ unitary _ vector , we see that @xmath861 , _ i.e. _ the twisted conjugation by @xmath794 using the grading automorphism @xmath862 is a reflection on the hyperplane perpendicular to @xmath794 . but reflections generate the orthogonal group @xmath863 so that the map @xmath232 , being defined for all invertible elements of the clifford algebra , when restricted to the subgroup generated by all products of an even number of unitary vectors ( @xmath864 ) , reduces to a homomorphism onto @xmath865 . this homomorphism may be restricted to @xmath866 . let us recall that there is a canonical vector space isomorphism between the clifford algebra and the exterior algebra , given explicitly in terms of an orthonormal basis by : @xmath867 direct computation then shows that @xmath868 coincides with @xmath869 and hence @xmath870 , the isomorphism being given by @xmath871\nonumber\end{aligned}\ ] ] let @xmath872 be an irreducible representation of @xmath848 . since @xmath868 is realized as a subspace of @xmath848 , we can compose @xmath742 with the isomorphism @xmath873 to obtain a representation , called the _ infinitesimal spin representation _ , of @xmath874 on @xmath146 . setting @xmath875 : @xmath876 _ i.e. _ , @xmath877 the infinitesimal spin representation satisfies the following basic relation : @xmath878=c(bv).\ ] ] comparing ( [ basic_rel ] ) with the identity from exercise [ ex:[dlambda a , v]=av ] , one realizes that @xmath857 and @xmath879 must be closely related . in order to express this relation , note that any real linear operator @xmath814 on @xmath789 can be written as @xmath880 , with @xmath881 linear in @xmath827 and @xmath882 antilinear in @xmath827 . for an element @xmath883 , @xmath884 is skewadjoint and @xmath885 antilinear and skewsymmetric ( in @xmath886 ) . then , it can be shown that for any @xmath883 , the following relation holds : @xmath887 where @xmath888 and @xmath889 ( with @xmath890 any orthonormal basis in @xmath827 )  @xcite . this equation plays an important role for the definition of the quantization map in the infinite dimensional case , discussed below .    in the infinite dimensional context , we begin with a separable real vector space @xmath789 on which a positive definite symmetric bilinear form @xmath508 is defined . it is assumed that @xmath789 is complete in the metric induced by @xmath508 , so that @xmath891 is a separable hilbert space , where @xmath892 . the clifford algebra is constructed as follows . consider the algebra @xmath893 obtained from the union of all algebras @xmath894 , where @xmath895 runs through all finite dimensional subspaces of @xmath789 . there is a unique trace @xmath896 on @xmath893 , inherited from the traces on each @xmath894 . the scalar product induced by the trace makes @xmath893 a prehilbert space . its completion @xmath897 allows one to represent @xmath893 as a subalgebra of @xmath898 via the gns construction . the closure of this algebra is then a @xmath0-algebra , which is defined to be the clifford algebra @xmath848 . the universal property remains valid in this context and in particular an orthogonal map @xmath899 extends to a @xmath0-algebra automorphism @xmath900 of @xmath848 ( bogoliubov automorphsim ) .    in order to construct a representation of the clifford algebra on fock space we need , as before , an orthogonal complex structure @xmath824 . the fock space @xmath842 is now defined as the completion of the exterior algebra @xmath841 with respect to the scalar product ( [ eq : scalar - prod ] ) . the action of @xmath848 on the exterior algebra is defined as in the finite dimensional case , its extension giving rise to an irreducible representation @xmath901 . as in the finite dimensional case , the action of the orthogonal group @xmath902 on the set of orthogonal complex structures is transitive , for given two complex structures @xmath824 and @xmath690 , there is a unitary map @xmath903 such that @xmath904 . as can be shown , if the bogoliubov automorphism @xmath900 is implementable by a unitary operator on fock space , then @xmath905 and @xmath906 are equivalent . in general however , they will be inequivalent . the converse also holds : implementability of @xmath900 follows from equivalence of the representations . necessary and sufficient conditions on @xmath509 for @xmath900 to be implementable are afforded by the shale - stinespring theorem ( _ cf . _  @xcite , thm . 6.16 ) , namely that the antilinear part @xmath907 of @xmath509 be hilbert - schmidt . in terms of the representations , it says that @xmath905 and @xmath906 are unitarily equivalent if and only if @xmath908 is hilbert - schimdt .      for the construction of quantized currents ( like the charge or the number operator ) we need a map taking self - adjoint operators on @xmath909 to self - adjoint operators on @xmath910 . the explicit construction of this map , which can be found in  @xcite , will not be presented here . instead , we will try to describe some of the motivations behind the construction . recall that @xmath902 acts transitively on the set of orthogonal complex structures , @xmath911 . since we have chosen a fixed complex structure @xmath824 to construct the representation @xmath905 on fock space , then any other complex structure will be given in terms of @xmath824 and of an element @xmath912 ( @xmath913 ) . recall also that an element of the unitary group @xmath914 is just an orthogonal map @xmath912 that commutes with @xmath824 . a unitary transformation @xmath915 is implemented in fock space by a map @xmath916 such that : @xmath917 now , the observables of the 1-particle theory correspond to elements of the lie algebra @xmath918 of the restricted orthogonal group , which is defined as @xmath919\ ; \mbox{is hilbert - schmidt}\rbrace$ ] . if an element @xmath920 is such that @xmath921=0 $ ] , then it follows that @xmath922 is self - adjoint on @xmath827 and it can be quantized by means of @xmath923 . the question of ( unitary ) equivalence of the quantization obtained by means of @xmath824 on one side and by means of @xmath913 on the other , may be formulated as follows . the complex structure @xmath924 is determined by @xmath509 . the map @xmath925 extends to an automorphism of the clifford algebra and the question is then whether this automorphism is implementable on fock space . so , given @xmath912 , we are looking for a unitary operator @xmath926 on fock space such that : @xmath927 in view of the inclusion @xmath928 , we see that we are looking for an extension of @xmath916 to @xmath902 . if @xmath929 $ ] is hilbert - schmidt , that is , if @xmath930 , then the map @xmath257 can be constructed ( it is the pin representation of @xmath931 on fock space ) . its restriction to @xmath932 is the spin representation , from which an infinitesimal version @xmath879 can be defined on a dense domain of fock space and turns out to be given by ( [ eq : mu = dlambda+tr ] ) without the trace term and , therefore , it is not a lie algebra homomorphism . this gives rise the so - called _ schwinger term _ , defined as @xmath933-\dot \mu ( [ a , b ] ) . \nonumber\ ] ] the schwinger term is a cocycle that gives an obstruction for @xmath934 to be a lie algebra homomorphism . as explained in  @xcite , the schwinger term is related to anomalies in quantum field theory and also to noncommutative geometry . it appears naturally when studying the virasoro algebra , and therefore is also related to the notion of _ central charge _ in conformal field theory . as we have seen , central charges do also appear when we study entanglement entropy in spin chains that display critical behavior . our aim in the next section is to give an idea of how all these concepts are interrelated . this will allow us to close the circle by providing a new topological interpretation of quantum criticality , for which the underlying geometry is not anymore the geometry of the space of external parameters , but a geometry in the sense of noncommutative geometry . this geometric interpretation of anomalies , central charges , schwinger cocycles _ etc . _ , has been known for several decades ( to the best of my knowledge the first person to realize this was araki  @xcite ) . but the connection to _ quantum critical phenomena _ does not seem to have been exploited . previously in these notes we discussed the gelfand - naimark theorem , which serves as a motivation for the notion of a noncommutative space . it turns out that the schwinger term described in the previous section also has a nice geometric interpretation , in the context of noncommutative geometry  @xcite . in order to give at least a glimpse of how this interpretation appears we will present , following  @xcite , a very basic ( yet instructive ) example , namely the gauss - bonnet theorem , in the language of noncommutative geometry . but before that we quickly review some aspects of the geometry of surfaces , following the beautiful presentation of the subject by pressley  @xcite . we thus start with a given 2-dimensional surface , described locally through a parametrization @xmath935 at each point , the tangent plane is generated by @xmath936 and @xmath937 and we can define the normal unit vector as @xmath938 the infinitesimal length element on the surface is given by @xmath939 where @xmath940 are the components of the _ first fundamental form _ : @xmath941 recall that the first and second fundamental forms enter the definition of gaussian curvature of a surface , which we now recall , starting from the curvature of a curve . for a plane curve , parametrized by @xmath942 with unit speed ( @xmath943 ) , let @xmath944 denote the unit normal at @xmath945 . then the ( signed ) curvature @xmath946 is defined through the relation @xmath947 . the curvature gives us an idea of how much the curve deviates from a straight line  @xcite . in fact , we may compute @xmath948    now we repeat the same reasoning , but with a surface . that is , we want to compute @xmath949 , where now @xmath810 denotes the unit normal to the surface . from a second order taylor expansion and using the fact that @xmath950 , we get @xmath951 where @xmath952 , @xmath953 and @xmath954 are the coefficients of the _ second fundamental form _ : @xmath955 now , for any smooth curve @xmath698 we have a decomposition of the form @xmath956 where @xmath957 and @xmath958 denote , respectively , the _ normal _ and _ geodesic _ curvatures of @xmath698 . to define the curvature of the surface we need to consider only curves with @xmath959 , for which the normal curvature takes the form @xmath960 the value of @xmath957 changes depending on the direction of the curve at the evaluation point , and it is clear it must attain maximum and minimum values . they are the _ principal curvatures _ , denoted as @xmath961 and @xmath962 . the gaussian curvature is defined as @xmath963 . we can obtain several useful formulas for the curvature using the fundamental forms . the principal curvatures @xmath961 and @xmath962 are obtained from an extremization problem , where we look for extremal values of the principal curvature as we change the direction of the curve -given by @xmath964- subject to the unit - speed restriction @xmath965 . so , with @xmath966 , @xmath967 , we want to extremize @xmath968 subject to the restriction @xmath969 . but we also have @xmath970    solve the extremization problem posed above , subject to the constraint @xmath969 , and show that the solution is given by the solution of the eigenvalue problem @xmath971 where the eigenvalue @xmath229 is the lagrange multiplier of the problem . show furthermore that , for a given solution @xmath972 , the corresponding normal curvature coincides precisely with @xmath229 and conclude that the gaussian curvature can be expressed as follows : @xmath973    there are other ways we can characterize @xmath690 . one of them uses the gauss map : for any point @xmath105 in the surface , consider the unit normal , @xmath974 . regarding this vector as a point in the unit sphere , we obtain a map @xmath975 from the surface to the unit sphere , the gauss map . as we go around a small loop centered at @xmath105 , the gauss map produces an image loop on the unit sphere . the quotient of the areas enclosed by the loops converges , in the limit where the original loop shrinks to a point , to the gaussian curvature . yet another way to obtain the gaussian curvature is by means of the _ weingarten map_. first notice that we have @xmath976 , @xmath977 and @xmath978 . this can be easily checked ; for instance , the first relation is obtained by taking the partial derivative of @xmath979 with respect to @xmath267 . the others are obtained in the same way . the weingarten map @xmath980 is most easily defined in the basis @xmath981 , where it has a matrix given by @xmath982 . in fact , if we define @xmath983 and write @xmath984 then it follows that @xmath985 now we compute @xmath986 from which the following identity is obtained : @xmath987 now , as the components of the unit normal @xmath988 are smooth functions of @xmath989 , we can compute their exterior derivatives @xmath990 in order to obtain @xmath991 this leads to the following formula for the euler characteristic : @xmath992 this formula is particularly well suited to explain the noncommutative viewpoint , to which we now turn , following the example presented by connes in  @xcite . let us consider an algebra @xmath158 ( without assuming other properties , for the moment being ) together with a trilinear map @xmath993 such that @xmath994 and @xmath995 now let @xmath996 be a ( continuous ) family of idempotents , @xmath997 . then it can be shown that the quantity @xmath998 does not depend on @xmath999 , providing an `` invariant '' of the algebra . a quick way to see what is going on is to assume that we can differentiate with respect to @xmath999 . in this case we have @xmath1000,\nonumber\ ] ] with @xmath1001 $ ] . using this fact and ( [ eq : cond(i ) ] ) , we obtain @xmath1002 , e_t , e_t).\nonumber\ ] ] but the last expression vanishes exactly . to see this , we put @xmath1003 and @xmath1004 in ( [ eq : cond(ii ) ] ) and use @xmath1005 . now , coming back to gauss - bonnet , if we take the algebra to be @xmath1006 , we see that the trilinear map @xmath1007 satisfies ( [ eq : cond(i ) ] ) and ( [ eq : cond(ii ) ] ) above , provided @xmath1008 . but , as explained in  @xcite , in this algebra there are no interesting idempotents . this can be remedied by adding a  bit of noncommutativity " to this algebra , that is , by replacing it by the following one : @xmath1009 now , recalling the gauss map , we realize that we can use the normal to the surface in order to obtain an idempotent , namely @xmath1010 where @xmath1011 , a `` vector '' whose components are the pauli matrices . then , if we define -following  @xcite- @xmath1012 we obtain ( an algebraic version of ) the euler characteristic . what we learn from the previous example is that it seems plausible to encode geometry and topology in algebraic structures related to a manifold @xmath167 . this is precisely the way noncommutative geometry works . after establishing such algebraic and operator - theoretic characterization of certain geometric or topological structures , one can try to keep the algebraic version , but now dropping the commutativity assumption . such is the case with differential calculus . let us then quickly list the main properties of de rham cohomology and see whether they can be formulated for more general algebras . given a compact manifold @xmath167 , the space of 1-forms @xmath1015 consists of elements of the form @xmath1016 , where @xmath59 and @xmath508 are smooth functions on @xmath167 , and @xmath1017 is the exterior derivative of @xmath508 . differential forms of order @xmath453 , @xmath1018 , are generated locally by expressions of the form @xmath1019 so that , in general , the exterior derivative @xmath1020 can be defined through @xmath1021 . the exterior derivative satisfies the leibniz rule @xmath1022 and fulfills the basic identity @xmath1023 , from which we obtain the ( de rham ) cohomology `` groups '' : @xmath1024 that contain important information about the topology of @xmath167 . finally , we also have stoke s theorem : @xmath1025 now suppose we are given a unital algebra @xmath158 . thinking of it as given by some space of functions on a `` virtual '' ( _ i.e. _ noncommutative ) space , we may try to define a differential calculus satisfying properties as close as possible to the ones recalled above . one possible construction goes as follows  @xcite :    in order to define the space of 1-forms , we consider the map @xmath1026 defined as @xmath1027 . its usefulness comes from the fact that it acts as a derivation : @xmath1028 but @xmath1029 is `` too big '' to be regarded as a space of 1-forms . we want to have a closer resemblance to @xmath1015 . recalling that 1-forms can always be put in the form @xmath1030 , we consider the restriction to the subspace of @xmath1029 generated by elements of the form @xmath1031 . noticing that @xmath1032 we arrive at the following definition : @xmath1033 where @xmath1034 is the multiplication map @xmath1035 . then , starting from @xmath1036 we can go on and define @xmath1037 for @xmath1038 . the exterior derivative , then , is simply given by @xmath1039 from these definitions we get , as can be easily checked :      this means that in any complex algebra we can find an analog of the space of differential forms ( _ i.e. _ , a graded differential algebra ) . the importance of this construction lies in the fact that it satisfies a universal property , as any derivation of @xmath158 into a bimodule factors through a bimodule morphism from @xmath1042 . we also want to have a notion of integration . in this context this is achieved through the introduction of _ cycles_. an @xmath810-dimensional cycle is a complex graded differential algebra ( @xmath1043 ) , along with an `` integral '' , _ i.e. _ a linear map @xmath1044 , such that : @xmath1045 the third requirement is tantamount to stokes theorem for the case of spaces without boundary . now , given an algebra @xmath158 , a _ cycle _ over @xmath158 is a cycle @xmath1046 together with a homomorphism @xmath1047  @xcite . a basic example is afforded by the de rham complex on a compact manifold without boundary .    given a cycle over @xmath158 we may now define , for @xmath1048 , @xmath1049 as shown in  @xcite , this map satisfies ( the generalization to @xmath810 of ) the relations ( [ eq : cond(i ) ] ) and ( [ eq : cond(ii ) ] ) , _ i.e. _ , it is a _ cyclic cocycle_.    given a complex algebra @xmath158 , an _ odd _ fredholm module over @xmath158 is given by a representation @xmath1050 on the algebra of bounded operators on a hilbert space , together with a self - adjoint operator @xmath1051 , such that @xmath1052 and such that @xmath1053 $ ] is a compact operator for each @xmath215 in @xmath158 . an _ even _ fredholm module contains , in addition to the structure of an odd fredholm module , an additional grading of @xmath21 , say @xmath698 , such that @xmath1054 and @xmath1055=0 , a\in \mathcal a$ ] .    on a fredholm module one can introduce the structure of a graded differential algebra defining the differential as @xmath1056.\ ] ] letting @xmath1057 be spanned by elements of the form @xmath1058 , and so on , we arrive at the formula @xmath1059_{\pm}$ ] as a suitable definition for @xmath1060 , where the bracket denotes a supercommutator ( depending on the degree of @xmath211 ) . an important class of examples for cyclic - cocycles is afforded by the so - called schwinger terms ( as defined in the previous section ) in quantum field theory . in the context of charged fermionic fields , we start with a real vector space @xmath789 ( given as a certain space of solutions to the classical field equation ) , a symmetric bilinear form @xmath508 on @xmath789 and a compatible ( orthogonal ) complex structure @xmath824 . physically , @xmath824 is determined by the spectrum of the hamiltonian . from the data @xmath1061 one then defines a complex hilbert space which is the complexification of @xmath789 using @xmath824 as the complex structure . so we get a hilbert space @xmath1062 , where the inner product is given by @xmath1063 we also have a representation of the field algebra on fock space , @xmath1064 now , the point is that the schwinger term @xmath1065 -\dot \mu[a , b ] = \frac{i}{8}\tr \left(j[j , a][j , b]\right),\ ] ] defines a cyclic 1-cocycle . a geometric interpretation of this cocycle is then made possible in the context of non - commutative geometry described above . when discussing the entanglement properties of spin chains , we found that the entanglement entropy of a sub - chain of length @xmath764 , at the critical point , scales logarithmically with @xmath764 . we mentioned that the constant in front of the logarithm was related to _ central charge _ of the corresponding conformal field theory . these central charges can also be understood as anomalies appearing in the quantization process of the effective field theories describing the spin chains . as an example , let us consider the ising chain : @xmath1066 using the wigner - jordan transformation and then defining the ( majorana ) operators  @xcite @xmath1067 we obtain @xmath1068 we can go to the continuum limit by assuming the sites in the chain are separated by a unit length @xmath1069 and defining @xmath1070 . then , to first order in @xmath1069 , we have : @xmath1071 defining the `` chiral '' components @xmath1072 and the corresponding spinor @xmath1073 one obtains , using heisenberg s equations of motion , @xmath1074 where @xmath1075 is such that @xmath1076 .      in the massless case , ( [ eq : dirac ] ) can be obtained from the lagrangian @xmath1080 . considering the spatial coordinate @xmath1081 to be defined on @xmath1082 ( which amounts to imposing periodic boundary conditions on the spin chain ) , expand the chiral fields in fourier modes and then express the hamiltonian density @xmath1083 , and the energy function @xmath1084 in terms of the fourier modes .    as the previous calculations show , when @xmath1078 the two chiral fields are completely decoupled from each other . thus , we may study them separately . let us consider @xmath1085 , for which the mode expansion must be of the form @xmath1086 imposing canonical anticommutation relations to the field ( which is real ) , we obtain : @xmath1087 from the previous exercise , we see that the contribution of @xmath1085 to the ( second quantized ) hamiltonian will be @xmath1088 where normal ordering has been introduced . now , how do we define normal ordering depends on what we regard as the `` vacuum state '' . this has important consequences for the commutation relations of the ( fourier components of the ) stress - energy tensor .    decomposing the stress - energy tensor @xmath1089 in light - cone components @xmath1090 and @xmath1091 , show that @xmath1092 , and that the fourier components ( @xmath1093 ) of @xmath1094 satisfy the following commutation relations ( virasoro algebra ) : @xmath1095 = ( m - n ) l_{m+n } + \frac{c}{12 } ( m^3-m)\delta_{m+n,0},\ ] ] with @xmath743 . the correct solution of the previous exercise depends crucially in a careful handling of the normal ordering prescription . this in turn is related to the `` filling - up of the dirac sea '' which , in mathematical terms , can be reduced to the problem of quantizing a linear system with an appropriately chosen complex structure . in fact , writing the hamiltonian density in terms of the spinor @xmath719 , we obtain @xmath1096 this means that the 1-particle hamiltonian ( which is defined on the hilbert space @xmath1097 ) is given by @xmath1098 . as we have seen , the splitting of the 1-particle hilbert space into positive and negative energy states gives rise to a complex structure , which is the one that should be used for quantization . when this is done , one obtains the anomalous term of the virasoro algebra as an anomaly ( schwinger term ) given precisely as a cyclic 1-cocyle of the form ( [ eq : schwinger - term ] ) . now , the relevance of this fact in the context of quantum phase transitions is that there are already some geometric characterizations of the critical point in terms of , _ e.g. _ , berry phases  @xcite . this has also been related to the behavior of certain chern numbers associated to the parameter space of the spin chain  @xcite . now , using araki s self - dual formalism , it should also be possible to compute the cocycle outside the critical point . the behavior of the cocycle as a function of the model s external parameters may provide a new geometric characterization of the critical point . h.  maassen , quantum probability and quantum information theory , in _ quantum information , computation and cryptography _ , eds . f.  benatti , m.  fannes , r.  floreanini and d.  petritis , lecture notes in physics , vol .  808 ( springer berlin heidelberg , 2010 ) pp . 65108 .      d  a guzmn , l  j uribe , a  valencia , f  j rodrguez , and l  quiroga , contrasting classical probability concepts with quantum mechanical behavior in the undergraduate laboratory , _ european journal of physics _ * 36 * ( 2015 ) , no .  5 , 055039 . e.  langmann , _ geometric methods for quantum field theory : proceedings of the summer school : villa de leyva , colombia , 12 - 30 july 1999 _ ( world scientific , 2001 ) , ch .  quantum theory of fermion systems : topics between physics and mathematics , pp . 170237 . h.  contreras and a.  reyes - lega , berry phases , quantum phase transitions and chern numbers , _ physica b : condensed matter _ * 403 * , 1301 ( 2008 ) , proceedings of the international conference on strongly correlated electron systems .
motivated by the sharp contrast between classical and quantum physics as probability theories , in these lecture notes i introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics . 
 aspects of the representation theory of c*-algebras will be motivated and illustrated in physical terms . 
 particular emphasis will be given to explicit examples from the theory of quantum phase transitions , where concepts coming from strands as diverse as quantum information theory , algebraic quantum physics and statistical mechanics agreeably converge , providing a more complete picture of the physical phenomena involved .