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 history for a system with a sum, or functional integral, over an infinity of possible histories to compute a quantum amplitude. Fig. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In physics, functional integration is integration over certain infinite-dimensional spaces. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ...

The path integral formulation 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. Richard Phillips Feynman (May 11, 1918 – February 15, 1988; IPA: ) was an American physicist known for expanding the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and particle theory. ... Doctor of Philosophy (Ph. ... John Archibald Wheeler (born July 9, 1911) is an eminent American theoretical physicist. ...

This formulation has proved crucial to the subsequent development of theoretical physics, since it provided the basis for the grand synthesis of the 1970s called the renormalization group which unified quantum field theory with statistical mechanics. If we realize that the Schrödinger equation is essentially a diffusion equation with an imaginary diffusion constant, then the path integral is a method for the enumeration of random walks. For this reason path integrals had also been used in the study of Brownian motion and diffusion before they were introduced in quantum mechanics. In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ... Quantum field theory (QFT) is the quantum theory of fields. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... This article or section does not cite any references or sources. ...

These are just three of the paths that contribute to the quantum amplitude for a particle moving from point A at some time t₀ to point B at some other time t₁.

Image File history File links Download high resolution version (821x800, 24 KB)These are just three of the paths included in the path integral used to calculate the quantum amplitude for a particle moving from point A to point B. File history Legend: (cur) = this is the current file, (del... Image File history File links Download high resolution version (821x800, 24 KB)These are just three of the paths included in the path integral used to calculate the quantum amplitude for a particle moving from point A to point B. File history Legend: (cur) = this is the current file, (del...

Formulating quantum mechanics

The path integral method is an alternative formulation of quantum mechanics. The canonical approach, pioneered by Erwin Schrödinger, Werner Heisenberg and Paul Dirac paid great attention to wave-particle duality and the resulting uncertainty principle by replacing Poisson brackets of classical mechanics by commutators between operators in quantum mechanics. The Hilbert space of quantum states and the superposition law of quantum amplitudes follows. The path integral starts from the superposition law, and exploits wave-particle duality to build a generating function for quantum amplitudes. Fig. ... Erwin Rudolf Josef Alexander Schrödinger (August 12, 1887 – January 4, 1961) was an Austrian physicist who achieved fame for his contributions to quantum mechanics, especially the Schrödinger equation, for which he received the Nobel Prize in 1933. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics, and acknowledged to be one of the most important physicists of the twentieth century. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ... In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ... In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... Quite literally, quantum state describes the state of a quantum system. ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ...

Abstract formulation

Feynman proposed the following postulates: This article or section does not adequately cite its references or sources. ...

1. The probability for any fundamental event is given by the square modulus of a complex amplitude.
2. The amplitude for some event is given by adding together all the histories which include that event.
3. The amplitude a certain history contributes is proportional to , where is reduced Planck's constant and S is the action of that history, or time integral of the Lagrangian.

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 includes them all. Not only that, it assigns all of them, no matter how bizarre, amplitudes of equal magnitude; only the phase, or argument of the complex number, varies. The contributions wildly different from the classical history are suppressed only by the interference of similar histories (see below). Probability is the chance that something is likely to happen or be the case. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In calculus, the integral of a function is an extension of the concept of a sum. ... This article is about a portion of a periodic process. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ...

It should be noted however, that the mathematical technique of path integrals, does not imply that real particles must actually follow the paths so constructed. Mathematical expansions of functions by other functions are a general technique, and as such the functions used are not required to have any physical interpretation at all. They are usually constructed for mathematical convenience, with no necessary analogy to the physical model that they are modeling. Indeed, in the case of the Feynman path integral, the integration is over imaginary time, so the relevance of the paths to the particle's real physical path is open to debate.

Feynman showed that his formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics. An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for the Hamiltonian corresponding to the given action. In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ...

Recovering the action principle

Feynman was initially attempting to make sense of a brief remark by Paul Dirac about the quantum equivalent of the action principle in classical mechanics. In the limit of action that is large compared to Planck's constant , the path integral is dominated by solutions which are stationary points of the action, since there the amplitudes of similar histories will tend to constructively interfere with one another. Conversely, for paths that are far from being stationary points of the action, the complex phase of the amplitude calculated according to postulate 3 will vary rapidly for similar paths, and amplitudes will tend to cancel. Therefore the important parts of the integral—the significant possibilities—in the limit of large action simply consist of solutions of the Euler-Lagrange equation, and classical mechanics is correctly recovered. Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... Stationary points (red pluses) and inflection points (green circles). ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... The Euler-Lagrange equation, developed by Leonhard Euler and Joseph-Louis Lagrange in the 1750s, is the major formula of the calculus of variations. ...

Action principles can seem puzzling to the student of physics because of their seemingly teleological quality: instead of predicting the future from initial conditions, one starts with a combination of initial conditions and final conditions and then finds the path in between, as if the system somehow knows where it's going to go. The path integral is one way of understanding why this works. The system doesn't have to know in advance where it's going; the path integral simply calculates the probability amplitude for a given process, and the stationary points of the action mark neighborhoods of the space of histories for which quantum-mechanical interference will yield large probabilities. Teleology (telos: end, purpose) is the philosophical study of design, purpose, directive principle, or finality in nature or human creations. ...

Concrete formulation

Feynman's postulates are somewhat ambiguous in that they do not define what an "event" is or the exact proportionality constant in postulate 3. The proportionality problem can be solved by simply normalizing the path integral by dividing the amplitude by the square root of the total probability for something to happen (resulting in that the total probability given by all the normalized amplitudes will be 1, as we would expect). Generally speaking one can simply define the "events" in an operational sense for any given experiment.

The equal magnitude of all amplitudes in the path integral tends to make it difficult to define it such that it converges and is mathematically tractable. For purposes of actual evaluation of quantities using path-integral methods, it is common to give the action an imaginary part in order to damp the wilder contributions to the integral, then take the limit of a real action at the end of the calculation. In quantum field theory this takes the form of Wick rotation. This article may be confusing for some readers, and should be edited to enhance clarity. ... In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ...

There is some difficulty in defining a measure over the space of paths. In particular, the measure is concentrated on "fractal-like" distributional paths. In mathematics, a measure is a function that assigns a number, e. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...

Time-slicing definition

For a particle in a smooth potential, the path integral is approximated by Feynman as the small-step limit over zig-zag paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position x₀ at time 0 to x_n at time t, the time interval can be divided up into n little segments of fixed duration Δt. This process is called time slicing. An approximation for the path integral can be computed as proportional to Look up zigzag in Wiktionary, the free dictionary. ...

where H is the entire history in which the particle zigzags from its initial to its final position linearly between all the values of

In the limit of n going to infinity, this becomes a functional integral. This limit does not, however, exist for the most important quantum-mechanical systems, the atoms, due to the singularity of the Coulomb potential at the origin. The problem was solved in 1979 by H. Duru and Hagen Kleinert (see here and here) by choosing Δt proportional to r and going to new coordinates whose square length is equal to r (Duru-Kleinert transformation). In physics, functional integration is integration over certain infinite-dimensional spaces. ... Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ... Hagen Kleinert, Photo taken in 2006 Hagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany, and Honorary Member of the Russian Academy of Creative Endeavors. ... 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 (singular like ). The Duru-Kleinert transformation replaces the diverging time-sliced path integral of Richard Feynman (which thus does not...

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 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. In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...

The connection with statistical mechanics follows. Considering only paths which begin and end in the same configuration, perform the Wick rotation t→it, 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 temperature . Strictly speaking, though, this is the partition function for a statistical field theory. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ... In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ... A statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. ...

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 Fig. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

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 In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ...

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. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Erwin Schrödinger, as depicted on the former Austrian 1000 Schilling bank note. ... The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ...

Quantum field theory

Today, the most common use of the path integral formulation is in quantum field theory. Quantum field theory (QFT) is the quantum theory of fields. ...

The propagator

A common use of the path integral is to calculate , a quantity (here written in bra-ket notation) known as the propagator. As such it is very useful in quantum field theory, where the propagator is an important component of Feynman diagrams. One way to do this, which Feynman used to explain photon and electron/positron propagators in quantum electrodynamics, is to apply the path integral to the motion of a single particle—one, however, that can roam back and forth through time as well as space in the course of its wanderings. (Such behavior can be reinterpreted as the contribution of the creation and annihilation of virtual particle-antiparticle pairs, so in this sense the single-particle restriction has already been loosened.) Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... In 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. ... Quantum field theory (QFT) is the quantum theory of fields. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, infrared radiation, microwaves, radio waves, and visible light are all forms of light. ... e- redirects here. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... In physics, a virtual particle is a particle which exists for such a short time and space that its energy and momentum do not have to obey the usual relationship. ... Corresponding to most kinds of particle, there is an associated antiparticle with the same mass and opposite charges. ...

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: 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. The magnitude of an electric field surrounding two equally charged (repelling) particles. ... In mathematics, the term functional is applied to certain functions. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ...

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. In physics, functional integration is integration over certain infinite-dimensional spaces. ...

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 (called a Wick rotation) makes the functional integral even more like a statistical partition function, and also tames some of the mathematical difficulties of working with these integrals. In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... 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: where S is the action functional. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ...

Expectation values

In quantum field theory, if the action is given by the functional of field configurations (which only depends locally on the fields), then the time ordered vacuum expectation value of polynomially bounded functional F, <F>, is given by Quantum field theory (QFT) is the quantum theory of fields. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... In mathematics, the term functional is applied to certain functions. ... In theoretical physics, path-ordering is the procedure (or a meta-operator ) of ordering a product of many operators according to the value of one chosen parameter: Here is a permutation that orders the parameters: Examples If an operator is not simply expressed as a product, but as a function... In quantum field theory the vacuum expectation value (also called condensate) of an operator is its average, expected value in the vacuum. ...

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.

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. The Euler-Lagrange equation, developed by Leonhard Euler and Joseph-Louis Lagrange in the 1750s, is the major formula of the calculus of variations. ... In mathematics and theoretical physics, the functional derivative is a generalization of the directional derivative. ... The Schwinger-Dyson equation, named after Julian Schwinger and Freeman Dyson, is an equation of quantum field theory (QFT). ...

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 rotation In quantum field theory, a nonlinear &#963; model is describes a scalar field &#931; which takes on values in a nonlinear manifold called the target manifold T. The target manifold is equipped with a Riemannian metric g. ... In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ...

which now becomes

for some H, goes to zero faster than any reciprocal of any polynomial for large values of φ, we can integrate by parts (after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger-Dyson equations: Look up reciprocal in Wiktionary, the free dictionary. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

for any polynomially bounded functional F.

in the deWitt notation. In physics, we often deal with classical models where the dynamical variables are a collection of functions {&#966;&#945;}&#945; over a d-dimensional space/spacetime manifold M where &#945; is the flavor index. ...

These equations are the analog of the on shell EL equations. In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ...

If J (called the source field) is an element of the dual space 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: In theoretical physics, a source field is a field whose multiple appears in the action, multiplied by the original field . ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In geometry, a translation slides an object by a vector a: Ta(p) = p + a. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

Note that

or

where

Basically, if is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of QFT, unlike its Wick rotated statistical mechanics analogue, because we have time ordering complications here!), then are its moments and Z is its Fourier transform. Quantum field theory (QFT) is the quantum theory of fields. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In theoretical physics, path-ordering is the procedure (or a meta-operator ) of ordering a product of many operators according to the value of one chosen parameter: Here is a permutation that orders the parameters: Examples If an operator is not simply expressed as a product, but as a function... -1... In mathematics, the Fourier transform is a certain linear operator that maps functions to other 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 In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...

and G is a functional of J, then

Then, from the properties of the functional integrals, we get the "master" Schwinger-Dyson equation: In physics, functional integration is integration over certain infinite-dimensional spaces. ... The Schwinger-Dyson equation, named after Julian Schwinger and Freeman Dyson, is an equation of quantum field theory (QFT). ...

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 quantum field theory, a nonlinear &#963; model is describes a scalar field &#931; which takes on values in a nonlinear manifold called the target manifold T. The target manifold is equipped with a Riemannian metric g. ... In quantum field theory, a nonlinear &#963; model is describes a scalar field &#931; which takes on values in a nonlinear manifold called the target manifold T. The target manifold is equipped with a Riemannian metric g. ...

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. As the degree of the Taylor series rises, it approaches the correct function. ... The Schwinger-Dyson equation, named after Julian Schwinger and Freeman Dyson, is an equation of quantum field theory (QFT). ...

Functional identity

If we perform a Wick rotation inside the functional integral, professors J. Garcia and Gerard 't Hooft showed using a functional differential equation that: In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ... Gerard t Hooft at Harvard University Gerardus (Gerard) t Hooft (born July 5, 1946) is a professor in theoretical physics at Utrecht University, The Netherlands. ...

where S is the Wick-rotated classical action of the particle,J is the classical action with an extra term "x" and delta here is the functional derivative operator

Ward-Takahashi identities

See main article Ward-Takahashi identity In quantum field theory a Ward-Takahashi identity is nowadays used to designate an identity between correlation functions that follows from symmetries, either global or gauged, of the theory, and which remains valid after renormalization. ...

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. In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ...

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 Lagrangian, and that for some function f where f only depends locally on φ (and possibly the spacetime position). Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

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. There are several meanings of derivation: A derivation in abstract algebra is a linear map that satisfies Leibniz law. ... There are several meanings of derivation: A derivation in abstract algebra is a linear map that satisfies Leibniz law. ... Possible alternative meanings of BRST are: BRST formalism Big Red Switch Time (or Big Red Switch Treatment): computer jargon for switching your computer off, when all other options for a more elegant shutdown have been exhausted. ... This article or section is in need of attention from an expert on the subject. ...

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) for more details. In physics, an anomaly is a classical symmetry — a symmetry of the Lagrangian — that is broken in quantum field theories. ...

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 principle), 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 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 (see the reference below) claim the interpretation explains the Einstein-Podolsky-Rosen paradox without resorting to nonlocality. It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ... This article or section is in need of attention from an expert on the subject. ... In quantum mechanics, the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. ... A physical theory is said to exhibit nonlocality if, in that theory, it is not possible to treat widely separated systems as independent. ...

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 decoherence is the general term for the consequences of irreversible quantum entanglement. ...

See also

• Theoretical and experimental justification for the Schrödinger equation

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. ...

Suggested reading

• Feynman, R. P., and Hibbs, A. R., Quantum Physics and Path Integrals, New York: McGraw-Hill, 1965 [ISBN 0-07-020650-3]. The historical reference, written by the Master himself and one of his students.
• Hagen Kleinert, 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, Jonh Wiley & Sons (New York-1981) [ISBN ]. The modern reference on the subject.
• 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, certainly the best 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.

Hagen Kleinert, Photo taken in 2006 Hagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany, and Honorary Member of the Russian Academy of Creative Endeavors. ... shelby was here 2004 (MMIV) was a leap year starting on Thursday of the Gregorian calendar. ...

Papers on-line

• Grosche, Christian  ; An Introduction into the Feynman Path Integral, lecture given at the graduate college Quantenfeldtheorie und deren Anwendung in der Elementarteilchen- und Festkörperphysik, Universität Leipzig, 16-26 November 1992. Full text available at : hep-th/9302097.
• MacKenzie, Richard ; Path Integral Methods and Applications, lectures given at Rencontres du Vietnam: VIth Vietnam School of Physics, Vung Tau, Vietnam, 27 December 1999 - 8 January 2000. Full text available at : quant-ph/0004090.
• DeWitt-Morette, Cécile ; Feynman's path integral - Definition without limiting procedure, Communication in Mathematical Physics 28(1) (1972) pp. 47–67. Full text available at : Euclide Project.
• Sukanya Sinha and Rafael D. Sorkin, "A Sum-over-histories Account of an EPR(B) Experiment", Found. of Phys. Lett. 4:303-335 (1991). Full text available at :"Sinha-Sorkin 1991".
• Cartier, Pierre & DeWitt-Morette, Cécile ; A new perspective on Functional Integration, Journal of Mathematical Physics 36 (1995) pp. 2137-2340. Full text available at : funct-an/9602005.