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in an early paper on the interaction between radiation and molecular dipole moments @xcite , dicke formulated a model which was later shown to exhibit a super - radiant phase transition@xcite . the notion that such phase transitions should exist in condensed matter systems has been investigated in a series of papers by preparata and coworkers@xcite and others@xcite . different workers have come to somewhat different conclusions concerning super - radiant phase transitions @xcite . some doubt has been expressed@xcite concerning the physical laboratory reality of super - radiant phase transition . the mathematical issues are as follows : ( i ) it appears,_at first glance _ , that quadratic terms ( in photon creation and annihilation operators ) enter into the model via quadratic terms in the vector potential @xmath0 . ( ii ) the quadratic terms in the `` corrected dicke model '' appear to destroy the super - radiant phase transition . our purpose is to show that if the dipole - field interaction is treated in a _ gauge invariant manner_@xcite then the interaction is _ strictly linear _ in the electric field @xmath1 . thus , quadratic terms are _ absent _ for purely electric dipole - photon interactions [ 28 ] . these considerations render likely the physical reality of condensed matter super - radiant phase transitions . to see what is involved , let us first consider a single molecule with an electric dipole moment @xmath2 given by @xmath3 where @xmath4 and @xmath5 denote respectively , the charge and position of the @xmath6 particle . suppose that the dipole moment interacts with a _ classical _ electric field which is uniform in space , but not in time @xmath7 the molecule will have a time dependent hamiltonian of the form @xmath8 where @xmath9 is the kinetic energy of @xmath10 particle in the molecule and @xmath11 is the internal coulomb energy of the molecule . employing the unitary operator @xmath12 in the time dependent canonical transformation @xmath13 yields @xmath14 for completeness of presentation , a derivation of eq.(5 ) is given in appendix a.    note : for classical electric fields of the form in eq.(2 ) , the hamiltonians in eqs.(3 ) and ( 6 ) are _ rigorously _ equivalent . furthermore , the hamiltonian in eq.(3 ) depends on linear and quadratic terms in the vector potential @xmath15 . on the other hand , the hamiltonian in eq.(6 ) depends only on the electric field @xmath16 and in a strictly _ gauge invariant _ and _ linear _ fashion .    in the work which follows , both the charged particles and the electromagnetic field will be treated using quantum mechanics . in sec.ii we discuss a single molecule in the presence of the quantized electromagnetic field . it is proved for the gauge invariant description of the dipole interactions that the coupling between the electromagnetic field and the molecule is linear in photon creation and annihilation operators the results for @xmath17 molecules are discussed in sec.iii . the resulting hamiltonian has four terms : ( i ) a sum of single molecule hamiltonians , ( ii ) a sum of screened coulomb interaction potentials between neighboring molecules , ( iii ) the radiation field energy and ( most importantly ) ( iv ) a _ strictly linear coupling _ between the dipole moments and the electric field . dicke - preparata models are defined and explored in sec.iv . in the concluding sec.v , the importance of the super - radiant phase transition associated with linear couplings will be discussed . the errors by some previous workers who falsely `` voided '' the super - radiant phase transition will be discussed in detail . consider , as in sec.i , a single molecule having a total number of @xmath18 charged particles with an electric dipole moment @xmath2 as in eq.(1 ) . the dipole is considered to interact with a quantized electromagnetic field ( photons ) @xmath19 and/or @xmath20 which is again uniform in space . the hamiltonian for the photons is given by @xmath21 where @xmath22 = \delta_{kk^\prime } .\ ] ] furthermore , for a `` quantization box '' of volume @xmath23 , @xmath24 and @xmath25 the matter hamiltonian for a single molecule in the electric dipole limit is then @xmath26    in total , the time independent hamiltonian for the molecule with dipole - quantized electromagnetic field interactions reads @xmath27 where @xmath28 is defined in eq.(7 ) , and @xmath29 is defined in eqs.(9 ) and ( 11 ) . in the quantum electrodynamic model , eq.(2 ) is replaced by the operator equation @xmath30\ ] ] leading to eq.(10 ) .    employing the unitary operator @xmath31 in the time - independent canonical transformation @xmath32 yields @xmath33 where the `` self energy '' contribution to the molecule is given by @xmath34 the derivation of eq.(16 ) is given in appendix b.    introducing the boson operators @xmath35 and the hermitian conjugate @xmath36 yields our final hamiltonian @xmath37 @xmath38 where @xmath39 and @xmath40 = \delta_{kk^\prime } .\ ] ] the central result of this section is the following :    * theorem : * for the electric dipole - electromagnetic field interaction , the resulting hamiltonian is that of free photons plus interaction terms linear in the photon creation and annihilation operators . _ quadratic interaction terms rigorously vanish . _    * proof : * see eqs.(18 ) and ( 19 ) , which were derived solely on the basis of the dipole interaction . let us now consider the case of @xmath17 molecules , each of which interact with the quantized electromagnetic field via the electric dipole moment . we consider the case of many molecules with electric dipole moments interacting with electromagnetic field modes whose minimum wavelength @xmath41 is large on the scale of the molecular size @xmath42 ; i.e. @xmath43 as shown in fig.1 below . the hamiltonian for @xmath17 molecules plus the electromagnetic field reads @xmath44 the radiation hamiltonian in the coulomb gauge @xmath45 is given by @xmath46 the internal @xmath47 molecular hamiltonian is given by @xmath48 @xmath49 where @xmath50 is the central position of the @xmath51 molecule . the internal hamiltonian of each molecule contains the internal coulomb potentials . in the last term on the right hand side of eq.(21 ) , @xmath52 describes the coulomb interactions between two different molecules . the long ranged part of this potential is of the conventional dipole - dipole interaction form @xmath53 where @xmath54 . the commutation relations of the quantum electrodynamic fields in the coulomb gauge are given by @xmath55= 4\pi i \hbar c { \bf \delta } ( { \bf r}-{\bf r}^\prime ) \ ] ] where the transverse delta function is defined as @xmath56 equivalently @xmath57 so that @xmath58 explicitly@xcite @xmath59 eqs.(25 ) and ( 30 ) imply @xmath60    the transverse distribution function @xmath61 removes the longitudinal part of a vector field . for example , if @xmath62 represents polarization , i.e. the molecular dipole moment per unit volume , then @xmath63 represents the transverse part of the polarization . now , consider the unitary operator @xmath64 eq.(26 ) , ( 33 ) and ( 34 ) imply @xmath65 so that the radiation hamiltonian of eq.(23 ) reads @xmath66 eqs.(17 ) , ( 23 ) , ( 31)-(33 ) and ( 36 ) imply that @xmath67 @xmath68 it is useful to define the `` screened '' ( or short ranged ) intermolecular coulomb potential with the long ranged dipole - dipole potential subtracted @xmath69 finally , the unitary transformation removes the vector potential from the internal degrees of freedom of a molecule as in the above sec.ii ; i.e. in the dipole interaction limit @xmath70    the `` grand finale '' of the algebraic ceremony of this section is the assertion that total transformed hamiltonian @xmath71 which for @xmath72 molecules takes the form @xmath73 @xmath74 where the internal molecular hamiltonian @xmath75 is independent of the transverse @xmath0 . the central eq.(41 ) of this section follows from eqs.(17 ) , ( 24 ) , ( 37)-(39 ) and ( 40 ) . the hamiltonian in eq.(41 ) has four parts : ( i ) a sum of intramolecular hamiltonians describing the internal kinetic and coulomb energies of each molecule , ( ii ) a sum of screened ( short ranged ) intermolecular coulomb potentials between neighboring molecules , ( iii ) the radiation field energy and ( iv ) a linear coupling between the dipole moment per unit volume and the electric field . within the dipole approximation , eq.(41 ) is exact but not yet amenable to rigorous mathematical solution . for the purpose of obtaining reasonable answers for ( say ) the free energy , further simplifications must be made . for the dicke - preparata model , the approximations are as follows : ( i ) the sum of intramolecular hamiltonians is modeled by a sum of `` two - level molecules '' , each described by pauli matrices @xmath76 ; @xmath77 ( ii ) the sum of screened intermolecular coulomb potentials between neighboring molecules is neglected @xmath78 ( iii ) the radiation field energy is modeled by a single photon oscillator mode @xmath79 ( iv ) the linear coupling between the dipole moment per unit volume and the electric field is modeled by the _ resonant _ photon creation and annihilation terms @xmath80 where @xmath81 and @xmath82 is the excitation matrix element of the electric dipole operator of a single molecule . putting the model replacement eqs.(43)-(46 ) into eq.(41 ) yields the dicke - preparata model @xmath83 where @xmath84 @xmath85 and @xmath86    the dicke - preparata model may be written in terms of the oscillator coordinate @xmath87 and momentum @xmath88 using @xmath89 yielding @xmath90 where the vector @xmath91 is given by @xmath92    the free energy @xmath93 of the model is determined by @xmath94 here , the trace over the oscillator degree of freedom may be taken as being classical with virtually zero error in the limit @xmath95 ; i.e. @xmath96 since @xmath97 and since eq.(53 ) implies @xmath98 it follows from eqs.(52 ) , ( 53 ) , and ( 55)-(57 ) that @xmath99 where @xmath100 the oscillator energy per molecule is @xmath101 and the number of molecules per unit volume @xmath102 is @xmath103    in the thermodynamic limit of eqs.(58)-(61 ) , the minimum free energy principle is that @xmath104 where @xmath105 and @xmath106    let @xmath107 represent the mean oscillator energy per molecule as a function of the molecular density @xmath108 and the temperature @xmath109 . if @xmath110 , then the system is in a super - radiant phase . if @xmath111 , then the system is in a normal ( incoherent radiation ) phase . the free energy per molecule @xmath112 is at a minimum for @xmath113 . setting @xmath114 yields the following implicit function equation for @xmath115 : @xmath116    the phase diagram in the @xmath117 plane follows from setting @xmath118 and solving for the critical temperature @xmath119 as a function of density . the super - radiant phase exists for @xmath120 and the normal phase exists for @xmath121 . the critical temperature @xmath122 where the critical density @xmath123 analytically , @xmath124 a plot of the phase diagram is shown below . the doubts about the physical reality of the super - radiant phases were discussed in great detail by i. bialynicki - birula and k. rzazewski@xcite , who came to the remarkable ( and dubious ) conclusion that the electric dipole - photon interaction could be gauged away . this incorrect conclusion was based on using a vector potential which is independent of both space and time . it is no wonder that such a pure `` gauge field '' had zero effect on the thermodynamic equations of state .    in reality , the transverse electric field in the coulomb gauge should be calculated using either @xmath125 or @xmath126 , \ \ { \rm ( schrodinger\ picture)}.\ ] ] the _ classical canonical transformation _ eq.(12 ) of i. bialynicki - birula and k. rzazewski@xcite ( in the dipole approximation ) should have been properly performed as in our eqs.(5 ) and ( 6 ) . the last term on the right hand side of both our eqs.(5 ) and ( 6 ) were incorrectly omitted in eq.(12 ) of i. bialynicki - birula and k. rzazewski@xcite . this incorrect classical canonical transformation eliminated the electric field @xmath127 and thereby the electric dipole - photon interaction . a proper procedure is to employ a fully quantum electrodynamic framework to obtain a non - trivial electric field @xmath127 _ before _ going to the classical electrodynamic oscillator trace evaluation . from this rigorous schrdinger picture point of view , eq.(12 ) of i. bialynicki - birula and k. rzazewski@xcite is still _ incorrect _ because eqs.(b2-b3 ) in our appendix b eqs.(b2-b3 ) were not taken into account in that work@xcite .    in any or all pictures , the electric dipole - photon interaction _ can not _ be described by gauging away the electric field @xmath20 . when the hamiltonian is expressed in terms of @xmath20 and @xmath128 , it is then properly gauge invariant and non - trivial changes _ very surely _ appear in the thermal equations of state . for example , the one - photon loop corrections to the equations of state have long been known to produce casimir - van der waals forces in condensed matter . such forces can not possibly be gauged away and i. bialynicki - birula and k. rzazewski@xcite are thereby in error . a further error is found in k. rzazewski , k. wodkiewicz and w. zakowicz@xcite , where it is asserted that the super - radiant phase transition is null and void due to quadratic coupling in the photon creation and annihilation operators . had the canonical transformation been carried out correctly , as in the present work , then it would have been noted that the coupling between the electric dipole moment and the photons in the gauge invariant eq.(41 ) is _ strictly linear_. the super - radiant phase transition remains intact . preparata has recently proposed several examples of super - radiant phases in a recent review@xcite . water is perhaps the most interesting example@xcite . in future work it would appear interesting to relate the super - radiant transition to an instability of the incoherent radiation phase which produces the conventional theoretical and experimental qed casimir effects . qed effects at the one photon loop level have long proved their importance in the understanding of the long ranged intermolecular forces@xcite . starting from the schrdinger equation @xmath129 one seeks a solution of the form @xmath130 where @xmath131 . putting eq.(a2 ) into eq.(a1 ) yields @xmath132 @xmath133 or equivalently @xmath134 where @xmath135 eq.(a5 ) is the required time dependent canonical transformation .      from eqs.(1 ) and ( 14 ) it follows that @xmath136 eqs.(b1 ) and ( 11 ) then imply @xmath137 thus , the vector potential in the dipole interaction theory may be `` gauged away '' . however , the dipole interaction returns to the full hamiltonian since eqs.(9 ) and ( 14 ) imply that @xmath138 and @xmath139 from eqs.(7 ) , ( b2 ) and ( b3 ) it follows that @xmath140 where the operator electric field @xmath127 is given in eq.(10 ) , and the molecular self energy @xmath141 is given in eq.(17 ) . thus the same canonical transformation which gauges away the vector potential in the molecule hamiltonian , brings back the electric dipole interaction @xmath142 when applied to the radiation hamiltonian via eq.(b4 ) . finally , from eqs.(b2 ) and ( b4 ) it follows that @xmath143 @xmath144 and eqs.(16 ) and ( 18 ) then follows from eq.(b5 ) . the proof of eq.(18 ) has been completed . dicke , _ phys . rev . _ * 93 * , 99 ( 1954 ) . k. hepp and e.h . lieb , _ phys . * a8 * , 2517 ( 1973 ) . k. hepp and e.h . lieb , _ ann . phys . _ * 76 * , 360 ( 1973 ) . e. del giudice , g. preparata and g. vitiello _ phys . lett . _ * 61 * , 1085 ( 1988 ) . g. preparata , `` quantum field theory of super radiance '' in _ problems of fundamental modern physics _ , r. cherubini , p. dalpiaz and b. minetti , world scientific , singapore ( 1990 ) . g. preparata , _ phys . _ * a38 * , 233 ( 1988 ) . enz , _ helv . . acta _ * 70 * , 141 ( 1997 ) . s. takeno and m. nagashima , _ prog . phys . _ * 57 * , 1507 ( 1977 ) . y.k . wong and f.t . hioe , _ phys . * a7 * , 831 ( 1973 ) . carmichel , c.w . gardiner and d.f . walls , _ phys . _ * a46 * , 47 ( 1973 ) . g. vertogem and a.s . de vires , _ phys . lett . _ * a48 * , 451 ( 1974 ) . hioe , _ phys . rev . _ * a8 * , 1440 ( 1973 ) . kudenko , a.p . slivinsky and g.m . zaslavsky , _ phys . lett . _ * a50 * , 411 ( 1975 ) . b.m . pimentel and a.h . lett . _ * a53 * , 200 ( 1975 ) zaslavsky , y.a . kudenko and a.p . slivinsky _ sov . phys . solid state _ * 16 * , 2197 ( 1975 ) . r. gilmore and c.m . bowden , _ phys . rev . _ * a13 * , 1898 ( 1976 ) . r. gilmore and c.m . * 17 * , 1617 ( 1976 ) . a.a.s . brito and w.f . wreszinski , _ helv . acta _ * 50 * , 373 ( 1977 ) . a. klemm and v.a . zagrebnov , _ physica _ * a36 * , 400 ( 1977 ) . k. rzazewski , k. wodkiewicz and w. zakowicz , _ phys . lett . _ * 35 * , 432 ( 1975 ) . k. rzazewski and k. wodkiewicz , _ phys . rev . _ * a13 * , 1967 ( 1976 ) . k. rzazewski , k. wodkiewicz and w. zakowicz , _ phys . * a58 * , 211 ( 1976 ) . i. bialynicki - birula and k. rzazewski , _ phys . _ * a19 * , 301 ( 1979 ) . k. gawedzki and k. rzazewski , _ phys . rev . _ * a23 * , 2134 ( 1981 ) . woolley , _ j. phys . lett . _ * a9 * , l15 ( 1976 ) . woolley , _ adv . phys . _ * 33 * , 153 ( 1975 ) . power and s. zienau , _ phil . _ * a251 * , 427 ( 1959 ) . g. preparata , e. del giudice and r. mele , _ mod . phys . lett . _ * b7 * , 1851 ( 1993 ) . d.p . craig and t. thirunamachandran , _ `` molecular quantum electrodynamics '' _ , p 56 , dover publications , new york ( 1998 ) . g. preparata , _ `` qed coherence in matter '' _ , world scientific , singapore ( 1995 ) . e. del giudice and g. preparata _ `` a new qed picture of water '' _ in _ `` macroscopic quantum coherence '' _ edited by e. sassaroli , y. srivastava , j. swain and a. widom , world scientific , singapore ( 1998 ) .
in a gauge invariant formulation of the molecular electric dipole - photon interaction , the rigorous coupling is strictly linear in the photon creation and photon annihilation operators . 
 the linear coupling allows for a super - radiant phase transition as in the hepp - lieb formulation . 
 a previous notion of a quadratic - coupling `` no - go theorem '' for super - radiance is incorrect . also incorrect 
 is a previous assertion that the dipole - photon coupling has absolutely no effect on the thermal equations of state . 
 these dubious assertions were based on incorrect canonical transformations which eliminated the electric field ( and thereby eliminated the dipole - photon interaction ) which is neither mathematically nor physically consistent . 
 the correct form of the canonical transformations are given in this work which allows for the physical reality of super - radiant condensed matter phases .