Path integral formulation
Encyclopedia
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 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 with 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 rather than a Hamiltonian
.
This formulation has proved 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 walk
s. For this reason path integrals were used in the study of Brownian motion
and diffusion
a while before they were introduced in quantum mechanics.
Recently path integrals have been expanded from Brownian paths to Lévy flight
s. The Lévy path integral formulation leads to fractional quantum mechanics
and a fractional Schrödinger equation
.
is the infinitesimal generator of time-translations. This means that the state at a slightly later time is related to the state at the current time by acting with the Hamiltonian operator (times −
). For states with a definite energy, this is a statement of the DeBroglie 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 considering special relativity
. The Hamiltonian tells you how to march forward in time, but the notion of 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 (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a Legendre transform, and the condition that determines the classical equations is that the Action is a minimum.
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 
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
. 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.
s 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:
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 histories of the system in between the initial and final states, including histories 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 would be correct to include histories in which the particle describes elaborate curlicues, histories in which the particle shoots off into outer space and flies back again, and so forth. The path integral assigns all of these histories amplitudes of equal magnitude but with varying phase
, or argument of the complex number
. The contributions that are wildly different from the classical history are suppressed only by the interference of similar, canceling histories (see below).
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.
Classical action principles are puzzling because of their seemingly teleological
quality: given a set of initial and final conditions one is able to find a unique path connecting them, as if the system somehow knows where it's going to end up and how it's going to get there. The path integral explains why this works in terms of quantum superposition. The system doesn't have to know in advance where it's going or what path it'll take: the path integral simply calculates the sum of the probability amplitudes for every possible path to any possible endpoint. After a long enough time, interference effects guarantee that only the contributions from the stationary points of the action give histories with appreciable probabilities.
at time 
to 
at time tb, the time sequence 
can be divided up into n little segments 
of fixed duration 
(the one remaining segment can be neglected, since finally the limit 
is considered), 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 
considered (see below), and 
corresponds to the position at the j-th time step, if the time integral is approximated by a sum of n terms.
(For a simplified, step by step, derivation of the above relation see Path Integrals in Quantum Theories: A Pedagogic 1st Step pdf vers)
In the limit of n going to infinity, this becomes a functional integral, which - apart from a nonessential factor - is directly the product of the probability amplitudes
- more precisely, since one must work with a continuous spectrum, the respective densities - to find the quantum mechanical particle at 
in the initial state xa and at tb in the final state xb.
Actually
is the classical Lagrangian
of the one-dimensional system considered,
, where 
is the Hamiltonian, with 
, 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
to 
with the integration measure 
is an arbitrary value of the interval corresponding to j, e.g. its center, 
.
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
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
.
):
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
is a description of quantum theory which generalizes the action principle
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. 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
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
.
The basic idea of the path integral formulation can be traced back to P. A. M. Dirac in his 1933 paper. The complete method was developed in 1948 by Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
. Some preliminaries were worked out earlier, in the course of his doctoral thesis work with John Archibald Wheeler
John Archibald Wheeler
John Archibald Wheeler was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission...
. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler-Feynman absorber theory using a Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
as a starting point rather than a Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
.
This formulation has proved crucial to the subsequent development of theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, 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
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
processes, and this provided the basis for the grand synthesis of the 1970s which unified quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
with the statistical field theory
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
of a fluctuating field near a second-order phase transition. The Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
of a method for summing up all possible random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
s. For this reason path integrals were used in the study of Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
and diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
a while before they were introduced in quantum mechanics.
Recently path integrals have been expanded from Brownian paths to Lévy flight
Lévy flight
A Lévy flight is a random walk in which the step-lengths have a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions...
s. The Lévy path integral formulation leads to fractional quantum mechanics
Fractional quantum mechanics
In physics, fractional quantum mechanics is a generalization of standard quantum mechanics. The term fractional quantum mechanics was coined by Nick Laskin...
and a fractional Schrödinger equation
Fractional Schrödinger equation
The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term fractional Schrödinger equation was coined by...
.
Quantum action principle
In ordinary quantum mechanics, the HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
is the infinitesimal generator of time-translations. This means that the state at a slightly later time is related to the state at the current time by acting with the Hamiltonian operator (times −
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
). For states with a definite energy, this is a statement of the DeBroglie relation between frequency and energy, and the general relation is consistent with that plus the superposition principle
Superposition principle
In physics and systems theory, the superposition principle , also known as superposition property, states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually...
.
But the Hamiltonian in classical mechanics is derived from a Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
, which is a more fundamental quantity considering special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
. The Hamiltonian tells you how to march forward in time, but the notion of 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 (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a Legendre transform, and the condition that determines the classical equations is that the Action is a minimum.
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 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
Quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...
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
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
.
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
. 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.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 relationCanonical commutation relation
In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:[x,p_x] = i\hbar...
s 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 probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
for an event is given by the squared length of a complex number called the "probability amplitude". - The probability amplitudeProbability amplitudeIn quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
is given by adding together the contributions of all the histories in configuration space. - The contribution of a history to the amplitude is proportional to
, where 
is reduced Planck's constant, and can be set equal to 1 by choice of units, while S is the actionAction (physics)In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
of that history, given by the time integral of the LagrangianLagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
along the corresponding path.
In order to find the overall probability amplitude for a given process, then, one adds up, or integrates
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, the amplitude of postulate 3 over the space of all possible histories of the system in between the initial and final states, including histories 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 would be correct to include histories in which the particle describes elaborate curlicues, histories in which the particle shoots off into outer space and flies back again, and so forth. The path integral assigns all of these histories amplitudes of equal magnitude but with varying phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
, or argument of the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
. The contributions that are wildly different from the classical history are suppressed only by the interference of similar, canceling histories (see below).
Feynman showed that this formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
, when the Hamiltonian is quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
for the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
corresponding to the given action.
Classical action principles are puzzling because of their seemingly teleological
Teleology
A teleology is any philosophical account which holds that final causes exist in nature, meaning that design and purpose analogous to that found in human actions are inherent also in the rest of nature. The word comes from the Greek τέλος, telos; root: τελε-, "end, purpose...
quality: given a set of initial and final conditions one is able to find a unique path connecting them, as if the system somehow knows where it's going to end up and how it's going to get there. The path integral explains why this works in terms of quantum superposition. The system doesn't have to know in advance where it's going or what path it'll take: the path integral simply calculates the sum of the probability amplitudes for every possible path to any possible endpoint. After a long enough time, interference effects guarantee that only the contributions from the stationary points of the action give histories with appreciable probabilities.
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 at time to at time tb, the time sequence can be divided up into n little segments of fixed duration (the one remaining segment can be neglected, since finally the limit is considered), 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 considered (see below), and corresponds to the position at the j-th time step, if the time integral is approximated by a sum of n terms.(For a simplified, step by step, derivation of the above relation see Path Integrals in Quantum Theories: A Pedagogic 1st Step pdf vers)
In the limit of n going to infinity, this becomes a functional integral, which - apart from a nonessential factor - is directly the product of the probability amplitudes
- more precisely, since one must work with a continuous spectrum, the respective densities - to find the quantum mechanical particle at in the initial state xa and at tb in the final state xb.Actually
is the classical LagrangianLagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
of the one-dimensional system considered,
, where is the Hamiltonian, with , 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
to with the integration measure is an arbitrary value of the interval corresponding to j, e.g. its center, .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
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 Duruİsmail Hakkı Duru
İsmail Hakkı Duru is a Turkish theoretical physicist and professor of Mathematics at the Izmir Institute of Technology where he is dean of the science faculty.-Scholarships:NATO scholarship during B.S...
and Hagen Kleinert
Hagen Kleinert
Hagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany , at theWest University of Timişoara, at thein Bishkek. He is also of the...
. 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
Duru-Kleinert transformation
The Duru-Kleinert transformation, named after İsmail Hakkı Duru and Hagen Kleinert, is a mathematical method for solving path integrals of physical systems with singular potentials, which is necessary for the solution of all atomic path integrals due to the presence of Coulomb potentials .The...
.
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 conceptually 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
. Each factor in the product is a Gaussian as a function of 
centered at x(t) with variance 
. The multiple integrals are a repeated convolutionConvolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of this Gaussian
with copies of itself at adjacent times.
-
Where the number of convolutions is
. 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 theoremCentral limit theoremIn probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
, 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 rotationWick rotationIn physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable...
. Then the same convolution argument as before gives the propagation kernel:-
Which, with the same normalization as before (not the sum-squares normalization! this function has a divergent norm), obeys a free Schrödinger equation-
This means that any superposition of K's will also obey the same equation, by linearity. Defining-
then
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
, 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
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 
they are additive; in any case one has with (1):
-
As mentioned, the spread in
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
and 
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
at each separate time is a separate integration variable. So it is legitimate to change variables in the integral by shifting: 
where 
is a different shift at each time but 
, 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
. 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
and 
, 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
by many orders of magnitude, we typically have destructive phase interference other than in the vicinity of those trajectories satisfying the Euler-Lagrange equationEuler-Lagrange equationIn calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...
, 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 there.
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
, so that:
-
-
This shows that the random walk is not differentiable, since the ratio that defines the derivative diverges with probability one.
The quantity
is ambiguous, with two possible meanings:
In ordinary calculus, the two are only different by an amount which goes to zero as
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
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 Ito lemma in stochastic calculus, and the (euclideanized) 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 (nonholonomic mapping explained here).
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 actionAction (physics)In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
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 mechanicsStatistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
follows. Considering only paths which begin and end in the same configuration, perform the Wick rotationWick rotationIn physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable...
, i.e., make time imaginary, and integrate over all possible beginning/ending configurations. The path integral now resembles the partition functionPartition function (statistical mechanics)Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
of statistical mechanics defined in a canonical ensemble with temperature
. Strictly speaking, though, this is the partition function for a statistical field theoryStatistical field theoryA statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. In other words, the microstates of the system are expressed through field configurations...
.
Clearly, such a deep analogy between quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
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 (imaginary) 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 (e.g. a particle moving in curved space) 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
. 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 theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
. 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 at the end of the day, 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 procedureRenormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
.
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 rulesOperator product expansion- 2D Euclidean quantum field theory :In quantum field theory, the operator product expansion is a Laurent series expansion of two operators...
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 failAnomaly (physics)In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics an anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking...
.
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 propagatorPropagatorIn quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...
. Superposing different values of the initial position
with an arbitrary initial state 
constructs the final state.
-
For a spatially homogenous system, where K(x, y) is only a function of (x-y), the integral is a convolutionConvolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
, 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 (x-y) 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
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 K(x-y,\tau) 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 theoremCentral limit theoremIn probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
, 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 
.
-
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 Euclidian propagator for a scalar particle. Rotating
to be imaginary gives the usual relativistic propagator, up to a -i 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
. 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 
, the dominant first term has the form:
-
This is the expression for the nonrelativistic Green's functionGreen's functionIn mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
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
of the first term must vanish, while the 
limit of the second term must vanish. In the fourier transform, this means shifting the pole in 
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
. 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
term introduces a small imaginary part to the 
, 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 fieldField (physics)In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
over all space. The action is referred to technically as a functionalFunctional (mathematics)In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
of the field:
where the field 
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 (not yet entirely successful) has been made toward making these functional integrals mathematically precise.
Such a functional integral is extremely similar to the partition functionPartition function (statistical mechanics)Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
in statistical mechanicsStatistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
. Indeed, it is sometimes called a partition functionPartition function (quantum field theory)In quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral:...
, and the two are essentially mathematically identical except for the factor of
in the exponent in Feynman's postulate 3. Analytically continuingAnalytic continuationIn complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
the integral to an imaginary time variable (called a Wick rotationWick rotationIn physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number 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 theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, if the actionAction (physics)In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
is given by the functionalFunctional (mathematics)In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
of field configurations (which only depends locally on the fields), then the time ordered vacuum expectation valueVacuum expectation valueIn quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
of polynomially bounded functional F, <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, we put the unadorned path integral in the denominator to normalize everything properly.
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 probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
which is written as P(B|A). In terms of path integration, since
this means:
where the functional
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 bangBig BangThe Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
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 equationEuler-Lagrange equationIn calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...
s as
(the left-hand side is a functional derivativeFunctional derivativeIn mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
; 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 equationSchwinger-Dyson equationThe Schwinger–Dyson equation , also Dyson-Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between Green functions in quantum field theories...
s.
If the functional measure
turns out to be translationally invariant (we'll assume this for the rest of this article, although this does not hold for, let's say nonlinear sigma models) and if we assume that after a Wick rotationWick rotationIn physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable...
which now becomes
for some H, goes to zero faster than any reciprocalReciprocal-In mathematics:*Multiplicative inverse, in mathematics, the number 1/x, which multiplied by x gives the product 1, also known as a reciprocal*Reciprocal rule, a technique in calculus for calculating derivatives of reciprocal functions...
of any polynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
for large values of φ, we can integrate by partsIntegration by partsIn calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
(after a Wick rotation, followed by a Wick rotation back) 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 (called the source field) is an element of the dual spaceDual spaceIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of the field configurations (which has at least an affine structure because of the assumption of the translational invariance for the functional measure), 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 (this shouldn't be taken too literally as an interpretation of QFTQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, unlike its Wick rotated statistical mechanicsStatistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
analogue, because we have time ordering complications here!), then
are its momentsMoment (mathematics)In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
and Z is its Fourier transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
.
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 seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
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 theoryLocal quantum field theoryThe Haag-Kastler axiomatic framework for quantum field theory, named after Rudolf Haag and Daniel Kastler, is an application to local quantum physics of C*-algebra theory. It is therefore also known as Algebraic 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.
Functional identity
If we perform a Wick rotationWick rotationIn physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable...
inside the functional integral, professors J. Garcia and Gerard 't Hooft showed using a functional differential equation, that
where S is the Wick-rotated classical action of the particle, J is the classical action with an extra term "x", delta (here) is the functional derivative operator and
Ward-Takahashi identities
- See main article Ward-Takahashi identity.
Now how about the on shell Noether's theoremNoether's theoremNoether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
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 (not local in the sense of a gauge symmetry, but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question). Let's also assume that the action is local in the sense that it is the integral over spacetime of a LagrangianLagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
, and that
for some function f where f only depends locally on φ (and possibly the spacetime position).
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 derivationDerivationDerivation may refer to:* Derivation , a function on an algebra which generalizes certain features of the derivative operator* Derivation * Derivation in differential algebra, a unary function satisfying the Leibniz product law...
which generates the one parameter group in question. We could have antiderivations as well, such as BRSTBRSTBRST may refer to:* BRST formalism and quantization in Yang-Mills theories* Big Red Switch Time , computer jargon for switching your computer off, when all other options for a more elegant shutdown have been exhausted...
and supersymmetrySupersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
.
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 (physics)Anomaly (physics)In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics an anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking...
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 (i.e. we are not insisting upon the gauge principleGauge principleIn physics, a gauge principle specifies a procedure for obtaining an interaction term from a free Lagrangian which is symmetric with respect to a continuous symmetry -- the results of localizing the global symmetry group must be accompanied by the inclusion of additional fields , with appropriate...
), 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 regulatorsRegularization (physics)-Introduction:In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator...
. Changing the scale of the regulator leads to the renormalization groupRenormalization groupIn theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
. 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 mechanicsInterpretation of quantum mechanicsAn interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations...
, 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 (see the reference below) claim the interpretation explains the Einstein-Podolsky-Rosen paradox without resorting to nonlocalityNonlocalityIn Classical physics, nonlocality is the direct influence of one object on another, distant object. In Quantum mechanics, nonlocality refers to the absence of a local, realist model in agreement with quantum mechanical predictions.Nonlocality may refer to:...
. (Note that the Copenhagen/pragmatism interpretation claims there is no paradox—only a sloppy materialism motivated question on the part of EPR—Joseph Wienberg a lecture. On the other hand, the fact that the EPR thought experiment (and its result) does represent the results of a QM experiment says that (despite the path dependence of parallelness/anti-parallelness in curved space) 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.
See also
- Theoretical and experimental justification for the Schrödinger equationTheoretical and experimental justification for the Schrödinger equationThe theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles...
- Static forces and virtual-particle exchangeStatic forces and virtual-particle exchangeStatic force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies...
- Feynman checkerboardFeynman checkerboardThe Feynman Checkerboard or Relativistic Chessboard model was Richard Feynman’s sum-over-paths formulation of the kernel for a free spin ½ particle moving in one spatial dimension...
- PropagatorPropagatorIn quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...
s - Wheeler–Feynman absorber theory
Suggested reading
- Feynman, R. P., and Hibbs, A. R., Quantum Mechanics and Path Integrals, New York: McGraw-Hill, 1965 [ISBN 0-07-020650-3]. The historical reference, written by the inventor of the path integral formulation himself and one of his students.
- Hagen KleinertHagen KleinertHagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany , at theWest University of Timişoara, at thein Bishkek. He is also of the...
, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore, 2004); Paperback ISBN 981-238-107-4 (also available online: PDF-files) - Zinn Justin, Jean ; Path Integrals in Quantum Mechanics, Oxford University Press (2004), [ISBN 0-19-856674-3]. A highly readable introduction to the subject.
- Schulman, Larry S. ; Techniques & Applications of Path Integration, John Wiley & Sons (New York-1981) [ISBN]. A modern reference on the subject.
- Ahmad, IshfaqIshfaq AhmadIshfaq Ahmad , D.Sc., Minister of State, SI, HI, NI, FPAS, is a Pakistani nuclear physicist, and well-known educationist and academic from Pakistan...
, ; Mathematical Integrals in Quantum Nature, The Nucleus (1971), pp 189–209, [ISBN] - Grosche, Christian & Steiner, Frank ; Handbook of Feynman Path Integrals, Springer Tracts in Modern Physics 145, Springer-Verlag (1998) [ISBN 3-540-57135-3]
- Ryder, Lewis H. ; Quantum Field Theory (Cambridge University Press, 1985), [ISBN 0-521-33859-X] Highly readable textbook; introduction to relativistic Q.F.T. for particle physics.
- Rivers, R.J. ; Path Integrals Methods in Quantum Field Theory, Cambridge University Press (1987) [ISBN 0-521-25979-7]
- Albeverio, S. & Hoegh-Krohn. R. ; Mathematical Theory of Feynman Path Integral, Lecture Notes in Mathematics 523, Springer-Verlag (1976) [ISBN].
- Glimm, James, and Jaffe, Arthur, Quantum Physics: A Functional Integral Point of View, New York: Springer-Verlag, 1981. [ISBN 0-387-90562-6].
- Gerald W. Johnson and Michel L. Lapidus ; The Feynman Integral and Feynman's Operational Calculus, Oxford Mathematical Monographs, Oxford University Press (2002) [ISBN 0-19-851572-3].
- Etingof, Pavel ; Geometry and Quantum Field Theory, M.I.T. OpenCourseWare (2002). This course, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals. A great introduction to Path Integrals (Chapter 1) and QFT in general., Universität Leipzig, 16–26 November 1992 |year=1992 |eprint=hep-th/9302097 }}
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