The path integral formulation of quantum
mechanics is a description of quantum
theory which generalizes the action
principle of classical mechanics. It
replaces the classical notion of a
single, unique trajectory for a system
with a sum, or functional integral, over
an infinity of possible trajectories to
compute a quantum amplitude.
The basic idea of the path integral
formulation can be traced back to
Norbert Wiener, who introduced the
Wiener integral for solving problems in
diffusion and Brownian motion. This idea
was extended to the use of the
Lagrangian in quantum mechanics by P. A.
M. Dirac in his 1933 paper. The complete
method was developed in 1948 by Richard
Feynman. Some preliminaries were worked
out earlier, in the course of his
doctoral thesis work by John Archibald
Wheeler. The original motivation stemmed
from the desire to obtain a
quantum-mechanical formulation for the
Wheeler–Feynman absorber theory using a
Lagrangian as a starting point.
This formulation has proven crucial to
the subsequent development of
theoretical physics, because it is
manifestly symmetric between time and
space. Unlike previous methods, the
path-integral allows a physicist to
easily change coordinates between very
different canonical descriptions of the
same quantum system.
The path integral also relates quantum
and stochastic processes, and this
provided the basis for the grand
synthesis of the 1970s which unified
quantum field theory with the
statistical field theory of a
fluctuating field near a second-order
phase transition. The Schrödinger
equation is a diffusion equation with an
imaginary diffusion constant, and the
path integral is an analytic
continuation of a method for summing up
all possible random walks. For this
reason path integrals were used in the
study of Brownian motion and diffusion a
while before they were introduced in
quantum mechanics.
Quantum action principle
In quantum mechanics, as in classical
mechanics, the Hamiltonian is the
generator of time-translations. This
means that the state at a slightly later
time differs from the state at the
current time by the result of acting
with the Hamiltonian operator. For
states with a definite energy, this is a
statement of the De Broglie relation
between frequency and energy, and the
general relation is consistent with that
plus the superposition principle.
But the Hamiltonian in classical
mechanics is derived from a Lagrangian,
which is a more fundamental quantity
relative to special relativity. The
Hamiltonian tells you how to march
forward in time, but the time is
different in different reference frames.
So the Hamiltonian is different in
different frames, and this type of
symmetry is not apparent in the original
formulation of quantum mechanics.
The Hamiltonian is a function of the
position and momentum at one time, and
it tells you the position and momentum a
little later. The Lagrangian is a
function of the position now and the
position a little later. The relation
between the two is by a Legendre
transform, and the condition that
determines the classical equations of
motion is that the action is an
extremum.
In quantum mechanics, the Legendre
transform is hard to interpret, because
the motion is not over a definite
trajectory. So what does the Legendre
transform mean? In classical mechanics,
with discretization in time,
and
where the partial derivative with
respect to  holds q(t + ε) fixed. The
inverse Legendre transform is:
where
and the partial derivative now is with
respect to p at fixed q.
In quantum mechanics, the state is a
superposition of different states with
different values of q, or different
values of p, and the quantities p and q
can be interpreted as noncommuting
operators. The operator p is only
definite on states that are indefinite
with respect to q. So consider two
states separated in time and act with
the operator corresponding to the
Lagrangian:
If the multiplications implicit in this
formula are reinterpreted as matrix
multiplications, what does this mean?
It can be given a meaning as follows:
The first factor is
If this is interpreted as doing a matrix
multiplication, the sum over all states
integrates over all q(t), and so it
takes the Fourier transform in q(t), to
change basis to p(t). That is the action
on the Hilbert space – change basis to p
at time t.
Next comes:
or evolve an infinitesimal time into the
future.
Finally, the last factor in this
interpretation is
which means change basis back to q at a
later time.
This is not very different from just
ordinary time evolution: the H factor
contains all the dynamical information –
it pushes the state forward in time. The
first part and the last part are just
doing Fourier transforms to change to a
pure q basis from an intermediate p
basis.
Another way of saying this is that since
the Hamiltonian is naturally a function
of p and q, exponentiating this quantity
and changing basis from p to q at each
step allows the matrix element of H to
be expressed as a simple function along
each path. This function is the quantum
analog of the classical action. This
observation is due to Paul Dirac.
Dirac further noted that one could
square the time-evolution operator in
the S representation
and this gives the time evolution
operator between time t and time t + 2ε.
While in the H representation the
quantity that is being summed over the
intermediate states is an obscure matrix
element, in the S representation it is
reinterpreted as a quantity associated
to the path. In the limit that one takes
a large power of this operator, one
reconstructs the full quantum evolution
between two states, the early one with a
fixed value of q(0) and the later one
with a fixed value of q(t). The result
is a sum over paths with a phase which
is the quantum action. Crucially, Dirac
identified in this paper the deep
quantum mechanical reason for the
principle of least action controlling
the classical limit.
Feynman's interpretation
Dirac's work did not provide a precise
prescription to calculate the sum over
paths, and he did not show that one
could recover the Schrödinger equation
or the canonical commutation relations
from this rule. This was done by
Feynman.
Feynman showed that Dirac's quantum
action was, for most cases of interest,
simply equal to the classical action,
appropriately discretized. This means
that the classical action is the phase
acquired by quantum evolution between
two fixed endpoints. He proposed to
recover all of quantum mechanics from
the following postulates:
The probability for an event is given by
the modulus length squared of a complex
number called the "probability
amplitude".
The probability amplitude is given by
adding together the contributions of all
paths in configuration space.
The contribution of a path is
proportional to , where S is the action
given by the time integral of the
Lagrangian along the path.
In order to find the overall probability
amplitude for a given process, then, one
adds up, or integrates, the amplitude of
postulate 3 over the space of all
possible paths of the system in between
the initial and final states, including
those that are absurd by classical
standards. In calculating the amplitude
for a single particle to go from one
place to another in a given time, it is
correct to include paths in which the
particle describes elaborate curlicues,
curves in which the particle shoots off
into outer space and flies back again,
and so forth. The path integral assigns
to all these amplitudes equal weight but
varying phase, or argument of the
complex number. Contributions from paths
wildly different from the classical
trajectory may be suppressed by
interference.
Feynman showed that this formulation of
quantum mechanics is equivalent to the
canonical approach to quantum mechanics
when the Hamiltonian is quadratic in the
momentum. An amplitude computed
according to Feynman's principles will
also obey the Schrödinger equation for
the Hamiltonian corresponding to the
given action.
The path integral formulation of quantum
field theory represents the transition
amplitude as a weighted sum of all
possible histories of the system from
the initial to the final state. And
Feynman diagram is a graphical
representation of a perturbative
contribution to the transition
amplitude.
Concrete formulation
Feynman's postulates can be interpreted
as follows:
= Time-slicing definition=
For a particle in a smooth potential,
the path integral is approximated by
zig-zag paths, which in one dimension is
a product of ordinary integrals. For the
motion of the particle from position xa
at time ta to xb at time tb, the time
sequence
can be divided up into n + 1 little
segments tj − tj − 1, where j = 1,...,n
+ 1, of fixed duration
This process is called time-slicing.
An approximation for the path integral
can be computed as proportional to
where  is the Lagrangian of the 1d
system with position variable x(t) and
velocity v = ẋ(t) considered, and dxj
corresponds to the position at the jth
time step, if the time integral is
approximated by a sum of n terms.
In the limit n → ∞, this becomes a
functional integral, which, apart from a
nonessential factor, is directly the
product of the probability amplitudes 
to find the quantum mechanical particle
at ta in the initial state xa and at tb
in the final state xb.
Actually  is the classical Lagrangian of
the one-dimensional system considered,
also
where  is the Hamiltonian,
, and the above-mentioned "zigzagging"
corresponds to the appearance of the
terms:
In the Riemannian sum approximating the
time integral, which are finally
integrated over x1 to xn with the
integration measure dx1...dxn, x̃j is an
arbitrary value of the interval
corresponding to j, e.g. its center,/2.
Thus, in contrast to classical
mechanics, not only does the stationary
path contribute, but actually all
virtual paths between the initial and
the final point also contribute.
Feynman's time-sliced approximation does
not, however, exist for the most
important quantum-mechanical path
integrals of atoms, due to the
singularity of the Coulomb potential
e2/r at the origin. Only after replacing
the time t by another path-dependent
pseudo-time parameter
the singularity is removed and a
time-sliced approximation exists, that
is exactly integrable, since it can be
made harmonic by a simple coordinate
transformation, as discovered in 1979 by
İsmail Hakkı Duru and Hagen Kleinert.
The combination of a path-dependent time
transformation and a coordinate
transformation is an important tool to
solve many path integrals and is called
generically the Duru–Kleinert
transformation.
= Free particle=
The path integral representation gives
the quantum amplitude to go from point x
to point y as an integral over all
paths. For a free particle action:
the integral can be evaluated
explicitly.
To do this, it is convenient to start
without the factor i in the exponential,
so that large deviations are suppressed
by small numbers, not by cancelling
oscillatory contributions.
Splitting the integral into time slices:
where the Dx is interpreted as a finite
collection of integrations at each
integer multiple of ε. Each factor in
the product is a Gaussian as a function
of x(t + ε) centered at x(t) with
variance ε. The multiple integrals are a
repeated convolution of this Gaussian Gε
with copies of itself at adjacent times.
Where the number of convolutions is T/ε.
The result is easy to evaluate by taking
the Fourier transform of both sides, so
that the convolutions become
multiplications.
The Fourier transform of the Gaussian G
is another Gaussian of reciprocal
variance:
and the result is:
The Fourier transform gives K, and it is
a Gaussian again with reciprocal
variance:
The proportionality constant is not
really determined by the time slicing
approach, only the ratio of values for
different endpoint choices is
determined. The proportionality constant
should be chosen to ensure that between
each two time-slices the time-evolution
is quantum-mechanically unitary, but a
more illuminating way to fix the
normalization is to consider the path
integral as a description of a
stochastic process.
The result has a probability
interpretation. The sum over all paths
of the exponential factor can be seen as
the sum over each path of the
probability of selecting that path. The
probability is the product over each
segment of the probability of selecting
that segment, so that each segment is
probabilistically independently chosen.
The fact that the answer is a Gaussian
spreading linearly in time is the
central limit theorem, which can be
interpreted as the first historical
evaluation of a statistical path
integral.
The probability interpretation gives a
natural normalization choice. The path
integral should be defined so that:
This condition normalizes the Gaussian,
and produces a Kernel which obeys the
diffusion equation:
For oscillatory path integrals, ones
with an i in the numerator, the
time-slicing produces convolved
Gaussians, just as before. Now, however,
the convolution product is marginally
singular since it requires careful
limits to evaluate the oscillating
integrals. To make the factors well
defined, the easiest way is to add a
small imaginary part to the time
increment . This is closely related to
Wick rotation. Then the same convolution
argument as before gives the propagation
kernel:
Which, with the same normalization as
before, obeys a free Schrödinger
equation
This means that any superposition of K's
will also obey the same equation, by
linearity. Defining
then ψt obeys the free Schrödinger
equation just as K does:
= The Schrödinger equation=
The path integral reproduces the
Schrödinger equation for the initial and
final state even when a potential is
present. This is easiest to see by
taking a path-integral over
infinitesimally separated times.
Since the time separation is
infinitesimal and the cancelling
oscillations become severe for large
values of ẋ, the path integral has most
weight for y close to x. In this case,
to lowest order the potential energy is
constant, and only the kinetic energy
contribution is nontrivial. The
exponential of the action is
The first term rotates the phase of ψ(x)
locally by an amount proportional to the
potential energy. The second term is the
free particle propagator, corresponding
to i times a diffusion process. To
lowest order in ε they are additive; in
any case one has with:
As mentioned, the spread in ψ is
diffusive from the free particle
propagation, with an extra infinitesimal
rotation in phase which slowly varies
from point to point from the potential:
and this is the Schrödinger equation.
Note that the normalization of the path
integral needs to be fixed in exactly
the same way as in the free particle
case. An arbitrary continuous potential
does not affect the normalization,
although singular potentials require
careful treatment.
= Equations of motion=
Since the states obey the Schrödinger
equation, the path integral must
reproduce the Heisenberg equations of
motion for the averages of x and ẋ
variables, but it is instructive to see
this directly. The direct approach shows
that the expectation values calculated
from the path integral reproduce the
usual ones of quantum mechanics.
Start by considering the path integral
with some fixed initial state
Now note that x(t) at each separate time
is a separate integration variable. So
it is legitimate to change variables in
the integral by shifting:  where ε(t) is
a different shift at each time but ε(0)
= ε(T) = 0, since the endpoints are not
integrated:
The change in the integral from the
shift is, to first infinitesimal order
in epsilon:
which, integrating by parts in t, gives:
But this was just a shift of integration
variables, which doesn't change the
value of the integral for any choice of
ε(t). The conclusion is that this first
order variation is zero for an arbitrary
initial state and at any arbitrary point
in time:
this is the Heisenberg equations of
motion.
If the action contains terms which
multiply ẋ and x, at the same moment in
time, the manipulations above are only
heuristic, because the multiplication
rules for these quantities is just as
noncommuting in the path integral as it
is in the operator formalism.
= Stationary phase approximation=
If the variation in the action exceeds ħ
by many orders of magnitude, we
typically have destructive phase
interference other than in the vicinity
of those trajectories satisfying the
Euler–Lagrange equation, which is now
reinterpreted as the condition for
constructive phase interference.
= Canonical commutation relations=
The formulation of the path integral
does not make it clear at first sight
that the quantities x and p do not
commute. In the path integral, these are
just integration variables and they have
no obvious ordering. Feynman discovered
that the non-commutativity is still
present.
To see this, consider the simplest path
integral, the brownian walk. This is not
yet quantum mechanics, so in the
path-integral the action is not
multiplied by i:
The quantity x(t) is fluctuating, and
the derivative is defined as the limit
of a discrete difference.
Note that the distance that a random
walk moves is proportional to √t, so
that:
This shows that the random walk is not
differentiable, since the ratio that
defines the derivative diverges with
probability one.
The quantity x ẋ is ambiguous, with two
possible meanings:
In elementary calculus, the two are only
different by an amount which goes to
zero as ε goes to zero. But in this
case, the difference between the two is
not zero:
give a name to the value of the
difference for any one random walk:
and note that f(t) is a rapidly
fluctuating statistical quantity, whose
average value is 1, i.e. a normalized
"Gaussian process". The fluctuations of
such a quantity can be described by a
statistical Lagrangian
and the equations of motion for f
derived from extremizing the action S
corresponding to  just set it equal to
1. In physics, such a quantity is "equal
to 1 as an operator identity". In
mathematics, it "weakly converges to 1".
In either case, it is 1 in any
expectation value, or when averaged over
any interval, or for all practical
purpose.
Defining the time order to be the
operator order:
This is called the Itō lemma in
stochastic calculus, and the canonical
commutation relations in physics.
For a general statistical action, a
similar argument shows that
and in quantum mechanics, the extra
imaginary unit in the action converts
this to the canonical commutation
relation,
= Particle in curved space=
For a particle in curved space the
kinetic term depends on the position and
the above time slicing cannot be
applied, this being a manifestation of
the notorious operator ordering problem
in Schrödinger quantum mechanics. One
may, however, solve this problem by
transforming the time-sliced flat-space
path integral to curved space using a
multivalued coordinate transformation.
= The path integral and the partition
function=
The path integral is just the
generalization of the integral above to
all quantum mechanical problems—
where  
is the action of the classical problem
in which one investigates the path
starting at time t=0 and ending at time
t = T, and Dx denotes integration over
all paths. In the classical limit, , the
path of minimum action dominates the
integral, because the phase of any path
away from this fluctuates rapidly and
different contributions cancel.
The connection with statistical
mechanics follows. Considering only
paths which begin and end in the same
configuration, perform the Wick rotation
, i.e., make time imaginary, and
integrate over all possible
beginning/ending configurations. The
path integral now resembles the
partition function of statistical
mechanics defined in a canonical
ensemble with inverse temperature
proportional to imaginary time, .
Strictly speaking, though, this is the
partition function for a statistical
field theory.
Clearly, such a deep analogy between
quantum mechanics and statistical
mechanics cannot be dependent on the
formulation. In the canonical
formulation, one sees that the unitary
evolution operator of a state is given
by
where the state α is evolved from time t
= 0. If one makes a Wick rotation here,
and finds the amplitude to go from any
state, back to the same state in time iT
is given by
which is precisely the partition
function of statistical mechanics for
the same system at temperature quoted
earlier. One aspect of this equivalence
was also known to Schrödinger who
remarked that the equation named after
him looked like the diffusion equation
after Wick rotation.
= Measure theoretic factors=
Sometimes we also have measure-theoretic
factors in the functional integral.
This factor is needed to restore
unitarity.
For instance, if
then it means that each spatial slice is
multiplied by the measure √g. This
measure can't be expressed as a
functional multiplying the  measure
because they belong to entirely
different classes.
Quantum field theory
The path integral formulation was very
important for the development of quantum
field theory. Both the Schrödinger and
Heisenberg approaches to quantum
mechanics single out time, and are not
in the spirit of relativity. For
example, the Heisenberg approach
requires that scalar field operators
obey the commutation relation
for x and y two simultaneous spatial
positions, and this is not a
relativistically invariant concept. The
results of a calculation are covariant,
but the symmetry is not apparent in
intermediate stages. If naive field
theory calculations did not produce
infinite answers in the continuum limit,
this would not have been such a big
problem – it would just have been a bad
choice of coordinates. But the lack of
symmetry means that the infinite
quantities must be cut off, and the bad
coordinates make it nearly impossible to
cut off the theory without spoiling the
symmetry. This makes it difficult to
extract the physical predictions, which
require a careful limiting procedure.
The problem of lost symmetry also
appears in classical mechanics, where
the Hamiltonian formulation also
superficially singles out time. The
Lagrangian formulation makes the
relativistic invariance apparent. In the
same way, the path integral is
manifestly relativistic. It reproduces
the Schrödinger equation, the Heisenberg
equations of motion, and the canonical
commutation relations and shows that
they are compatible with relativity. It
extends the Heisenberg type operator
algebra to operator product rules which
are new relations difficult to see in
the old formalism.
Further, different choices of canonical
variables lead to very different seeming
formulations of the same theory. The
transformations between the variables
can be very complicated, but the path
integral makes them into reasonably
straightforward changes of integration
variables. For these reasons, the
Feynman path integral has made earlier
formalisms largely obsolete.
The price of a path integral
representation is that the unitarity of
a theory is no longer self-evident, but
it can be proven by changing variables
to some canonical representation. The
path integral itself also deals with
larger mathematical spaces than is
usual, which requires more careful
mathematics not all of which has been
fully worked out. The path integral
historically was not immediately
accepted, partly because it took many
years to incorporate fermions properly.
This required physicists to invent an
entirely new mathematical object – the
Grassmann variable – which also allowed
changes of variables to be done
naturally, as well as allowing
constrained quantization.
The integration variables in the path
integral are subtly non-commuting. The
value of the product of two field
operators at what looks like the same
point depends on how the two points are
ordered in space and time. This makes
some naive identities fail.
= The propagator=
In relativistic theories, there is both
a particle and field representation for
every theory. The field representation
is a sum over all field configurations,
and the particle representation is a sum
over different particle paths.
The nonrelativistic formulation is
traditionally given in terms of particle
paths, not fields. There, the path
integral in the usual variables, with
fixed boundary conditions, gives the
probability amplitude for a particle to
go from point x to point y in time T.
This is called the propagator.
Superposing different values of the
initial position  with an arbitrary
initial state  constructs the final
state.
For a spatially homogeneous system,
where K(x, y) is only a function of, the
integral is a convolution, the final
state is the initial state convolved
with the propagator.
For a free particle of mass m, the
propagator can be evaluated either
explicitly from the path integral or by
noting that the Schrödinger equation is
a diffusion equation in imaginary time
and the solution must be a normalized
Gaussian:
Taking the Fourier transform in produces
another Gaussian:
and in p-space the proportionality
factor here is constant in time, as will
be verified in a moment. The Fourier
transform in time, extending K(p; T) to
be zero for negative times, gives the
Green's Function, or the frequency space
propagator:
Which is the reciprocal of the operator
which annihilates the wavefunction in
the Schrödinger equation, which wouldn't
have come out right if the
proportionality factor weren't constant
in the p-space representation.
The infinitesimal term in the
denominator is a small positive number
which guarantees that the inverse
Fourier transform in E will be nonzero
only for future times. For past times,
the inverse Fourier transform contour
closes toward values of E where there is
no singularity. This guarantees that K
propagates the particle into the future
and is the reason for the subscript on
G. The infinitesimal term can be
interpreted as an infinitesimal rotation
toward imaginary time.
It is also possible to reexpress the
nonrelativistic time evolution in terms
of propagators which go toward the past,
since the Schrödinger equation is
time-reversible. The past propagator is
the same as the future propagator except
for the obvious difference that it
vanishes in the future, and in the
gaussian t is replaced by. In this case,
the interpretation is that these are the
quantities to convolve the final
wavefunction so as to get the initial
wavefunction.
Given the nearly identical only change
is the sign of E and ε. The parameter E
in the Green's function can either be
the energy if the paths are going toward
the future, or the negative of the
energy if the paths are going toward the
past.
For a nonrelativistic theory, the time
as measured along the path of a moving
particle and the time as measured by an
outside observer are the same. In
relativity, this is no longer true. For
a relativistic theory the propagator
should be defined as the sum over all
paths which travel between two points in
a fixed proper time, as measured along
the path. These paths describe the
trajectory of a particle in space and in
time.
The integral above is not trivial to
interpret, because of the square root.
Fortunately, there is a heuristic trick.
The sum is over the relativistic
arclength of the path of an oscillating
quantity, and like the nonrelativistic
path integral should be interpreted as
slightly rotated into imaginary time.
The function  can be evaluated when the
sum is over paths in Euclidean space.
This describes a sum over all paths of
length  of the exponential of minus the
length. This can be given a probability
interpretation. The sum over all paths
is a probability average over a path
constructed step by step. The total
number of steps is proportional to , and
each step is less likely the longer it
is. By the central limit theorem, the
result of many independent steps is a
Gaussian of variance proportional to .
The usual definition of the relativistic
propagator only asks for the amplitude
is to travel from x to y, after summing
over all the possible proper times it
could take.
Where  is a weight factor, the relative
importance of paths of different proper
time. By the translation symmetry in
proper time, this weight can only be an
exponential factor, and can be absorbed
into the constant α.
This is the Schwinger representation.
Taking a Fourier transform over the
variable can be done for each value of 
separately, and because each separate 
contribution is a Gaussian, gives whose
Fourier transform is another Gaussian
with reciprocal width. So in p-space,
the propagator can be reexpressed
simply:
Which is the Euclidean propagator for a
scalar particle. Rotating p0 to be
imaginary gives the usual relativistic
propagator, up to a factor of and an
ambiguity which will be clarified below.
This expression can be interpreted in
the nonrelativistic limit, where it is
convenient to split it by partial
fractions:
For states where one nonrelativistic
particle is present, the initial
wavefunction has a frequency
distribution concentrated near p0 = m.
When convolving with the propagator,
which in p space just means multiplying
by the propagator, the second term is
suppressed and the first term is
enhanced. For frequencies near p0 = m,
the dominant first term has the form:
This is the expression for the
nonrelativistic Green's function of a
free Schrödinger particle.
The second term has a nonrelativistic
limit also, but this limit is
concentrated on frequencies which are
negative. The second pole is dominated
by contributions from paths where the
proper time and the coordinate time are
ticking in an opposite sense, which
means that the second term is to be
interpreted as the antiparticle. The
nonrelativistic analysis shows that with
this form the antiparticle still has
positive energy.
The proper way to express this
mathematically is that, adding a small
suppression factor in proper time, the
limit where t → −∞ of the first term
must vanish, while the t → +∞ limit of
the second term must vanish. In the
Fourier transform, this means shifting
the pole in p0 slightly, so that the
inverse Fourier transform will pick up a
small decay factor in one of the time
directions:
Without these terms, the pole
contribution could not be unambiguously
evaluated when taking the inverse
Fourier transform of p0. The terms can
be recombined:
Which when factored, produces opposite
sign infinitesimal terms in each factor.
This is the mathematically precise form
of the relativistic particle propagator,
free of any ambiguities. The ε term
introduces a small imaginary part to the
α = m2, which in the Minkowski version
is a small exponential suppression of
long paths.
So in the relativistic case, the Feynman
path-integral representation of the
propagator includes paths which go
backwards in time, which describe
antiparticles. The paths which
contribute to the relativistic
propagator go forward and backwards in
time, and the interpretation of this is
that the amplitude for a free particle
to travel between two points includes
amplitudes for the particle to fluctuate
into an antiparticle, travel back in
time, then forward again.
Unlike the nonrelativistic case, it is
impossible to produce a relativistic
theory of local particle propagation
without including antiparticles. All
local differential operators have
inverses which are nonzero outside the
lightcone, meaning that it is impossible
to keep a particle from travelling
faster than light. Such a particle
cannot have a Greens function which is
only nonzero in the future in a
relativistically invariant theory.
= Functionals of fields=
However, the path integral formulation
is also extremely important in direct
application to quantum field theory, in
which the "paths" or histories being
considered are not the motions of a
single particle, but the possible time
evolutions of a field over all space.
The action is referred to technically as
a functional of the field: S[ϕ] where
the field ϕ(xμ) is itself a function of
space and time, and the square brackets
are a reminder that the action depends
on all the field's values everywhere,
not just some particular value. In
principle, one integrates Feynman's
amplitude over the class of all possible
combinations of values that the field
could have anywhere in space–time.
Much of the formal study of QFT is
devoted to the properties of the
resulting functional integral, and much
effort has been made toward making these
functional integrals mathematically
precise.
Such a functional integral is extremely
similar to the partition function in
statistical mechanics. Indeed, it is
sometimes called a partition function,
and the two are essentially
mathematically identical except for the
factor of i in the exponent in Feynman's
postulate 3. Analytically continuing the
integral to an imaginary time variable
makes the functional integral even more
like a statistical partition function,
and also tames some of the mathematical
difficulties of working with these
integrals.
= Expectation values=
In quantum field theory, if the action
is given by the functional  of field
configurations, then the time ordered
vacuum expectation value of polynomially
bounded functional F, , is given by
The symbol  here is a concise way to
represent the infinite-dimensional
integral over all possible field
configurations on all of space–time. As
stated above, the unadorned path
integral in the denominator ensures
proper normalization.
= As a probability=
Strictly speaking the only question that
can be asked in physics is: "What
fraction of states satisfying condition
A also satisfy condition B?" The answer
to this is a number between 0 and 1
which can be interpreted as a
probability which is written as P(B|A).
In terms of path integration, since 
this means:
where the functional Oin[ϕ] is the
superposition of all incoming states
that could lead to the states we are
interested in. In particular this could
be a state corresponding to the state of
the Universe just after the big bang
although for actual calculation this can
be simplified using heuristic methods.
Since this expression is a quotient of
path integrals it is naturally
normalised.
= Schwinger–Dyson equations=
Since this formulation of quantum
mechanics is analogous to classical
action principles, one might expect that
identities concerning the action in
classical mechanics would have quantum
counterparts derivable from a functional
integral. This is often the case.
In the language of functional analysis,
we can write the Euler–Lagrange
equations as
(the left-hand side is a functional
derivative; the equation means that the
action is stationary under small changes
in the field configuration). The quantum
analogues of these equations are called
the Schwinger–Dyson equations.
If the functional measure  turns out to
be translationally invariant and if we
assume that after a Wick rotation
which now becomes
for some H, goes to zero faster than a
reciprocal of any polynomial for large
values of φ, we can integrate by parts
to get the following Schwinger–Dyson
equations for the expectation:
for any polynomially bounded functional
F.
in the deWitt notation.
These equations are the analog of the on
shell EL equations.
If J is an element of the dual space of
the field configurations, then, the
generating functional Z of the source
fields is defined to be:
Note that
or
where
Basically, if  is viewed as a functional
distribution, then  are its moments and
Z is its Fourier transform.
If F is a functional of φ, then for an
operator K, F[K] is defined to be the
operator which substitutes K for φ. For
example, if
and G is a functional of J, then
Then, from the properties of the
functional integrals
we get the "master" Schwinger–Dyson
equation:
or
If the functional measure is not
translationally invariant, it might be
possible to express it as the product 
where M is a functional and  is a
translationally invariant measure. This
is true, for example, for nonlinear
sigma models where the target space is
diffeomorphic to Rn. However, if the
target manifold is some topologically
nontrivial space, the concept of a
translation does not even make any
sense.
In that case, we would have to replace
the  in this equation by another
functional 
If we expand this equation as a Taylor
series about J = 0, we get the entire
set of Schwinger–Dyson equations.
Localization
The path integrals are usually thought
of as being the sum of all paths through
an infinite space–time. However, in
Local quantum field theory we would
restrict everything to lie within a
finite causally complete region, for
example inside a double light-cone. This
gives a more mathematically precise and
physically rigorous definition of
quantum field theory.
= Ward–Takahashi identities=
See main article Ward–Takahashi
identity.
Now how about the on shell Noether's
theorem for the classical case? Does it
have a quantum analog as well? Yes, but
with a caveat. The functional measure
would have to be invariant under the one
parameter group of symmetry
transformation as well.
Let's just assume for simplicity here
that the symmetry in question is local.
Let's also assume that the action is
local in the sense that it is the
integral over spacetime of a Lagrangian,
and that  for some function f where f
only depends locally on φ.
If we don't assume any special boundary
conditions, this would not be a "true"
symmetry in the true sense of the term
in general unless f=0 or something.
Here, Q is a derivation which generates
the one parameter group in question. We
could have antiderivations as well, such
as BRST and supersymmetry.
Let's also assume  for any polynomially
bounded functional F. This property is
called the invariance of the measure.
And this does not hold in general. See
anomaly for more details.
Then,
which implies
where the integral is over the boundary.
This is the quantum analog of Noether's
theorem.
Now, let's assume even further that Q is
a local integral
where
so that
where
(this is assuming the Lagrangian only
depends on φ and its first partial
derivatives! More general Lagrangians
would require a modification to this
definition!). Note that we're NOT
insisting that q(x) is the generator of
a symmetry, but just that Q is. And we
also assume the even stronger assumption
that the functional measure is locally
invariant:
Then, we would have
Alternatively,
The above two equations are the
Ward–Takahashi identities.
Now for the case where f=0, we can
forget about all the boundary conditions
and locality assumptions. We'd simply
have
Alternatively,
The need for regulators and
renormalization
Path integrals as they are defined here
require the introduction of regulators.
Changing the scale of the regulator
leads to the renormalization group. In
fact, renormalization is the major
obstruction to making path integrals
well-defined.
The path integral in quantum-mechanical
interpretation
In one philosophical interpretation of
quantum mechanics, the "sum over
histories" interpretation, the path
integral is taken to be fundamental and
reality is viewed as a single
indistinguishable "class" of paths which
all share the same events. For this
interpretation, it is crucial to
understand what exactly an event is. The
sum over histories method gives
identical results to canonical quantum
mechanics, and Sinha and Sorkin claim
the interpretation explains the
Einstein–Podolsky–Rosen paradox without
resorting to nonlocality. does represent
the results of a QM experiment says that
all contributions of paths close to
black holes cancel in the action for an
EPR style experiment here on earth.)
Some advocates of interpretations of
quantum mechanics emphasizing
decoherence have attempted to make more
rigorous the notion of extracting a
classical-like "coarse-grained" history
from the space of all possible
histories.
Quantum gravity
Whereas in quantum mechanics the path
integral formulation is fully equivalent
to other formulations, it may be that it
can be extended to quantum gravity,
which would make it different from the
Hilbert space model. Feynman had some
success in this direction and his work
has been extended by Hawking and others.
Approaches that use this method include
causal dynamical triangulations and
spinfoam models.
Quantum tunneling
Quantum tunnelling can be modeled by
using the path integral formation to
determine the action of the trajectory
through a potential barrier. Using the
WKB approximation, the tunneling rate
can be determined to be of the form
with the effective action  and
pre-exponential factor . This form is
specifically useful in a dissipative
system, in which the systems and
surroundings must be modeled together.
Using the Langevin equation to model
Brownian motion, the path integral
formation can be used to determine an
effective action and pre-exponential
model to see the effect of dissipation
on tunnelling. From this model,
tunneling rates of macroscopic systems
can be predicted.
See also
Theoretical and experimental
justification for the Schrödinger
equation
Static forces and virtual-particle
exchange
Feynman checkerboard
Propagators
Wheeler–Feynman absorber theory
Feynman–Kac formula
References
Notes
Suggested reading
Feynman, R. P. and Hibbs, A. R.. Quantum
Mechanics and Path Integrals. New York:
McGraw-Hill. ISBN 0-07-020650-3.  The
historical reference, written by the
inventor of the path integral
formulation himself and one of his
students.
Hagen Kleinert. Path Integrals in
Quantum Mechanics, Statistics, Polymer
Physics, and Financial Markets.
Singapore: World Scientific. ISBN
981-238-107-4. 
Zinn Justin, Jean. Path Integrals in
Quantum Mechanics. Oxford University
Press. ISBN 0-19-856674-3.  A highly
readable introduction to the subject.
Schulman, Larry S.. Techniques &
Applications of Path Integration. New
York: John Wiley & Sons. ISBN
0-486-44528-3.  A modern reference on
the subject.
Ahmad, Ishfaq. Mathematical Integrals in
Quantum Nature. The Nucleus. pp.
189–209. 
Inomata, Akira, Kuratsuji, Hiroshi, and
Gerry, Christopher. Path Integrals and
Coherent States of SU(2) and SU(1,1).
Singapore: World Scientific. ISBN
981-02-0656-9. 
Grosche, Christian & Steiner, Frank.
Handbook of Feynman Path Integrals.
Springer Tracts in Modern Physics 145.
Springer-Verlag. ISBN 3-540-57135-3. 
Tomé, Wolfgang A.. Path Integrals on
Group Manifolds. Singapore: World
Scientific. ISBN 981-02-3355-8. 
Discusses the definition of Path
Integrals for systems whose kinematical
variables are the generators of a real
separable, connected Lie group with
irreducible, square integrable
representations.
Klauder, John R.. A Modern Approach to
Functional Integration. New York:
Birkhäuser. ISBN 978-0-8176-4790-2. 
Ryder, Lewis H.. Quantum Field Theory.
Cambridge University Press. ISBN
0-521-33859-X.  Highly readable
textbook; introduction to relativistic
QFT for particle physics.
Rivers, R.J.. Path Integrals Methods in
Quantum Field Theory. Cambridge
University Press. ISBN 0-521-25979-7. 
Albeverio, S. & Hoegh-Krohn. R..
Mathematical Theory of Feynman Path
Integral. Lecture Notes in Mathematics
523. Springer-Verlag. ISBN
0-387-07785-5. 
Glimm, James, and Jaffe, Arthur. Quantum
Physics: A Functional Integral Point of
View. New York: Springer-Verlag. ISBN
0-387-90562-6. 
Simon, Barry. Functional Integration and
Quantum Phyiscs. New York: Academic
Press. ISBN 0821869418.  A
mathematically rigorous introduction to
Functional Integration
Gerald W. Johnson and Michel L. Lapidus.
The Feynman Integral and Feynman's
Operational Calculus. Oxford
Mathematical Monographs. Oxford
University Press. ISBN 0-19-851572-3. 
Etingof, Pavel. "Geometry and Quantum
Field Theory". MIT OpenCourseWare.  This
course, designed for mathematicians, is
a rigorous introduction to perturbative
quantum field theory, using the language
of functional integrals.
Zee, Anthony. Quantum Field Theory in a
Nutshell. Princeton University Press.
ISBN 978-0-691-14034-6.  A great
introduction to Path Integrals and QFT
in general.
Grosche, Christian. "An Introduction
into the Feynman Path Integral".
arXiv:hep-th/9302097. 
MacKenzie, Richard. "Path Integral
Methods and Applications".
arXiv:quant-ph/0004090. 
DeWitt-Morette, Cécile. "Feynman's path
integral: Definition without limiting
procedure". Communication in
Mathematical Physics 28: 47–67.
Bibcode:1972CMaPh..28...47D.
doi:10.1007/BF02099371. MR 0309456. 
Sinha, Sukanya; Sorkin, Rafael D.. "A
Sum-over-histories Account of an EPR(B)
Experiment". Foundations of Physics
Letters 4: 303–335.
Bibcode:1991FoPhL...4..303S.
doi:10.1007/BF00665892. 
Cartier, Pierre; DeWitt-Morette, Cécile.
"A new perspective on Functional
Integration". Journal of Mathematical
Physics 36: 2137–2340.
arXiv:funct-an/9602005.
Bibcode:1995JMP....36.2237C.
doi:10.1063/1.531039. 
External links
Path integral on Scholarpedia
Path Integrals in Quantum Theories: A
Pedagogic 1st Step
