This article is about a formulation of quantum mechanics. For integrals along a path, also known as line or contour integrals, see Path integral. This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...

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.

Richard Feynman developed the path integral formulation of quantum mechanics in 1948 (some preliminaries were worked out earlier, in the course of his doctoral thesis work with John Archibald Wheeler) as a description of quantum theory corresponding to 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. Richard Phillips Feynman (May 11, 1918–February 15, 1988) (surname pronounced FINE-man; in IPA) was one of the most influential American physicists of the 20th century, expanding greatly the theory of quantum electrodynamics. ... Fig. ... Doctor of Philosophy (Ph. ... John Archibald Wheeler (born 1911) is an American theoretical physicist. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... Classical mechanics is a model of the physics of forces acting upon bodies. ... In physics, functional integration is integration over certain infinite-dimensional spaces. ... In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ...

Path integrals have also been used in the study of Brownian motion and diffusion. An example of 1000 steps of Brownian motion in two dimensions. ... Diffusion is the spontaneous spreading of something such as particles, heat, or momentum. ...

Postulates

Feynman proposed the following postulates: For the algebra software named Axiom, see Axiom computer algebra system. ...

1. The probability for any fundamental event is given by the absolute square 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 I(H) 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 size; 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). The word probability derives from the Latin probare (to prove, or to test). ... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a functional of the dynamical variables which concisely describes the equations of motion of the system. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... Phase, from the Greek phasis, meaning appearance, has a number of related meanings in English. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ...

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 widely observed fact about nature. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ... In physics, Hamiltonian has distinct but closely related meanings. ...

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. In physics, functional integration is integration over certain infinite-dimensional spaces. ... 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 "fractallike" distributional paths. In mathematics, a measure is a function that assigns a number, e. ... The Mandelbrot set, named after its discoverer, is a famous example of a fractal. ... This page deals with mathematical distributions. ...

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, (August 8, 1902 - October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ... In mathematics, particularly in calculus, a stationary point is a point on the graph of a function where the tangent to the graph is parallel to the x-axis or, equivalently, where the derivative of the function equals zero (known as a critical number). ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ...

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 is the philosophical study of purpose (from the Greek teleos, perfect, complete, which in turn comes from telos, end, result). ...

Example: Single-particle mechanics

In the case of the motion of a particle, the path integral can be formally thought of as the small-step limit of an integral over zig-zags: for instance, for one-dimensional motion of a particle from position x₀ at time 0 to x_n at time t, the time interval can be divided up into little segments of duration Δt and the path integral can be computed as proportional to Zig-Zag is a company that makes products related to tobacco consumption. ...

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

x_j = x(jΔt).

In the limit, this is an integral over an infinite-dimensional space—a functional integral. In physics, functional integration is integration over certain infinite-dimensional spaces. ...

The path integral in quantum field theory

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

The propagator

A common use of the path integral is to calculate <q1,t1|q0,t0>, 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 application of quantum mechanics to fields. ... A Feynman diagram is a bookkeeping device for performing calculations in quantum field theory, invented by American physicist Richard Feynman. ... In physics, the photon (from Greek φοτος, meaning light) is a quantum of excitation of the quantised electromagnetic field and is one of the elementary particles studied by quantum electrodynamics (QED) which is the oldest part of the Standard Model of particle physics. ... Properties The electron (sometimes called negatron; commonly represented as e−) is a subatomic particle. ... A positron is the antiparticle of the electron. ... Quantum electrodynamics (QED) is a quantum field theory of electromagnetism. ... In the description of the interaction between elementary particles in quantum field theory, a virtual particle is a temporary elementary particle, used to describe an intermediate stage in the interaction. ... For each kind of particle, there is an associated antiparticle with the same mass but opposite electromagnetic, weak, and strong charges, as well as spin. ...

Functionals of fields

However, the path-integral formulation is also extremely important in direct application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a field over all space. The action is referred to technically as a functional of the field: S[φ] where the field φ(x^μ) 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. A field is an open land area, used for growing agricultural crops. ... 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 statistics, 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 S 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 application of quantum mechanics to fields. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... 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... The vacuum expectation value (also called vacuum 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. In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... In mathematics and theoretical physics, the functional derivative is a generalization of the usual derivative that arises in the calculus of variations. ... 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 σ model is describes a scalar field Σ 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. ...

e^iS[φ],

which now becomes

e ^{− H[φ]}

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: In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... 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 {φα}α over a d-dimensional space/spacetime manifold M where α 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 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). ...

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. QFT can represent Quantitative Feedback Theory. ... Statistical mechanics is the application of statistics, 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... See also moment (physics). ... The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

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 This article is about operators in mathematics, for other kinds of operators see operator (disambiguation). ...

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 σ model is describes a scalar field Σ 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 S 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). ...

Ward-Takahashi identities

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 expresses the one-to-one correspondence between the symmetries and the conservation laws. ...

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 functional of the dynamical variables which 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. ... In particle physics, supersymmetry is a hypothetical symmetry that relates bosons and fermions. ...

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, the word anomaly is used to describe a classical symmetry—i. ...

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 q(x)[φ(y)]=δ^(d)(x-y)Q[φ(y)] 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're NOT insisting upon the gauge principle), but just that Q is. And let's also assume the even stronger assumption that the functional measure is locally invariant:

.

Then, we'd 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 claim the interpretation explains the Einstein-Podolsky-Rosen paradox without resorting to nonlocality. In a nontechnical sense, an interpretation of quantum mechanics is an attempt to answer the question: what exactly is quantum mechanics talking about? Quantum mechanics has been very successful in predicting experimental results. ... 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. ... Nonlocality in quantum mechanics, refers to the property of entangled quantum states in which both the entangled states collapse simultaneously upon measurement of one of their entangled components, regardless of the spatial separation of the two states. ...

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. ...

References

• Feynman, R. P., and Hibbs, A. R., Quantum Physics and Path Integrals, New York: McGraw-Hill, 1965. ISBN 0-070-20650-3
• Glimm, James, and Jaffe, Arthur, Quantum Physics: A Functional Integral Point of View, New York: Springer-Verlag, 1981. ISBN 0-387-90562-6
• Pokorski, Stefan, Gauge Field Theories, Cambridge: Cambridge University Press, 1987. ISBN 0-521-36846-4
• Sakurai, J. J., Modern Quantum Mechanics, Tuan, San Fu, ed. Redwood City, California: Addison-Wesley, 1985. ISBN 0-8053-7501-5
• Sinha, Sukanya and Sorkin, Rafael Dolnick. "A Sum-over-histories Account of an EPR(B) Experiment". Foundations of Physics Letters, 4:303-335, 1991. (also available online: PostScript (http://physics.syr.edu/~sorkin/some.papers/63.eprb.ps))

• 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 (http://www.physik.fu-berlin.de/~kleinert/b5))