Functional integral

Functional integral (path integral, path integral, Feynman path integral, Feynman integral) is a record or result of functional integration (path integration). Finds the greatest application in quantum physics ( quantum field theory, string theory , etc.) and statistical physics, as well as in the study of a number of classes of stochastic processes in general.

Functional integration formally means calculating the integral of some functional Φ over the space of functions x ( t ) or some subset ^{[1] of} such a space:

\int Dx\,\Phi [x],

{\ displaystyle \ int Dx \, \ Phi [x],}

{\ displaystyle \ int Dx \, \ Phi [x],}

which is defined as the limit of the (finite-dimensional) integral over the space of certain finite-dimensional approximations of the functions x ( t ) as the dimension of these approximations tends to infinity; the usual and easiest way is to consider the function x on a finite set of points $t_{1},t_{2},\dots ,t_{N}$ ${\ displaystyle t_ {1}, t_ {2}, \ dots, t_ {N}}$ ${\ displaystyle t_ {1}, t_ {2}, \ dots, t_ {N}}$ , then determining the functional integral in the simplest case of a uniform partition, which can be limited to, as

\lim _{N\to \infty }\int \int \dots \int dx_{1}\,dx_{2}\dots dx_{N}\,\Phi [x_{1},x_{2},\dots ,x_{N}],

{\ displaystyle \ lim _ {N \ to \ infty} \ int \ int \ dots \ int dx_ {1} \, dx_ {2} \ dots dx_ {N} \, \ Phi [x_ {1}, x_ {2 }, \ dots, x_ {N}],}

{\ displaystyle \ lim _ {N \ to \ infty} \ int \ int \ dots \ int dx_ {1} \, dx_ {2} \ dots dx_ {N} \, \ Phi [x_ {1}, x_ {2 }, \ dots, x_ {N}],}

where under $\Phi [x_{1},x_{2},\dots ,x_{N}]$ ${\ displaystyle \ Phi [x_ {1}, x_ {2}, \ dots, x_ {N}]}$ ${\ displaystyle \ Phi [x_ {1}, x_ {2}, \ dots, x_ {N}]}$ I mean the corresponding approximation of the functional Φ [ x ], integration is implied separately by $x_{1},x_{2},\dots ,x_{N}$ ${\ displaystyle x_ {1}, x_ {2}, \ dots, x_ {N}}$ ${\ displaystyle x_ {1}, x_ {2}, \ dots, x_ {N}}$ from $-\infty$ ${\ displaystyle - \ infty}$ $- \ infty$ before $+\infty$ ${\ displaystyle + \ infty}$ $+ \ infty$ (in case of fixed $x_{1}$ ${\ displaystyle x_ {1}}$ $x_ {1}$ and $x_{N}$ ${\ displaystyle x_ {N}}$ $x_ {N}$ you do not need to integrate over them).

The correctness of this definition is already called into question in the sense that it has not been proved even for many of those cases that are of physical interest, not to mention a more general formulation of the question, the very existence of the limit (in particular, its identity when choosing different types of partitions; moreover, in a number of examples, different types give different results) and in many cases there is no way to specify clear criteria for choosing the “right” type of partition that will lead to the desired result, which means that the measure is correctly defined integration has not been proven even for many of those cases that are of physical interest, at least in the usual sense.

Another serious difficulty is the exact calculation of such integrals (with the exception of the Gaussian case).

Nevertheless, even the fact that at least integrals of the Gaussian type are accurately calculated gives a lot for the application of the method of functional integration. In particular, this result can be taken as the definition of the functional integral for this case and prove that, being so defined, it really has the properties of an integral: it allows integration by parts, change of variables, etc. ^[2]

The physical meaning of the functional integral usually reduces to calculating the sum (superposition) of a certain quantity (usually this is the probability for classical statistics or the probability amplitude for quantum mechanics) along “all” trajectories (that is, along all available classical particles in the case of Brownian motion and over to all imaginable in the case of quantum mechanics).

Content

Primary Application

Models

The usual random walk is capable of generating, when reformulating, the path integral with a certain action. This is generally relatively obvious in simple cases.

It was shown that a similar method of generating a continuum integral with the usual action also works in the two-dimensional case - to obtain the action for the string (two-dimensional object, taking into account the time dimension).

Physical Analogies

The analogue of the path integral for a point particle is the statistical sum (statistical weight) for a polymer thread ^[3] .

Calculation

Accurate calculation

As mentioned above, the exact calculation of a functional integral of the form

\int Dx\,e^{kS[x]},

{\ displaystyle \ int Dx \, e ^ {kS [x]},}

where k can be purely imaginary in the quantum case or real in the case of classical diffusion, is possible only in the case when it is of the Gaussian type, that is, when the action S is quadratic in x (the Lagrangian is quadratic in x and its derivatives, or maybe , even in some similar cases: the main thing is that S be a quadratic form, in the real case it is negative definite).

The method comes down to writing a discrete version, as defined at the beginning of the article. Then the (ordinary) integrals in the formula are exactly taken (as Gaussian ), and then we can go to the limit.

Rough Calculation

Numerical methods

Computational methods associated with finding the values of path integrals using computers, including quadrature formulas such as Simpson formulas and other methods, have been developed quite extensively by 2010, although they are used mainly by narrow specialists and for the most part are not known to physicists.

History

The first appearance of path integrals seems to relate to the work of Einstein and Smoluchowski ^{[ clarify ]} on the theory of Brownian motion .

The fundamentals of the mathematical theory of such integrals are connected with the works of Wiener of the 1920s . However, their rigorous and fairly complete mathematical theory still encounters significant difficulties (related to the question of the correct introduction of measures on the space of functions, to the problem of proving the independence of the limit on the type of partition in a fairly general case).

In 1933 (in the work “Lagrangian in quantum mechanics”) Dirac proposed the idea of using the path integral in quantum mechanics.

Feynman implemented this program in the late 1940s by developing a formal integral of continuum, which proved to be extremely fruitful in theoretical physics. This meant the emergence of a technically new (which had - besides purely technical - besides a number of intuitive advantages) a method for constructing quantum theories, which later became almost the most popular among theoreticians. Already on the basis of the formalism of the path integral, Feynman himself constructed such a basic technique of quantum field theory as Feynman diagrams .

Using the continuum integral, fundamental results were obtained, such as, for example, the proof of the renormalizability of the Yang - Mills theory ( Faddeev and Popov ).

Notes

↑ The most typical example of an integration domain in a function space is the set of all functions of a given space that satisfy the condition of fixing their values at two points (at the ends of the segment).
↑ Article in the Physical Encyclopedia (A. A. Slavnov).
↑ Polyakov, 1999 .

Literature

Simon B. Functional Integration and Quantum Physics. - Academic Press, 1979.
Berezin F.A. Second quantization method. - M .: Nauka , 1986 .-- 320 p.
Zinn-Justin J. Continuous integral in quantum mechanics. - M .: Fizmatlit , 2010 .-- 360 p.
Lobanov Yu. Yu. Methods of approximate functional integration for the numerical study of models in quantum physics (dissertation for the degree of Doctor of Physics and Mathematics). - M. , 2009.
Polyakov A.M. Gauge Fields and Strings. - Izhevsk: RHD, 1999 .-- 316 p.
Popov V. N. Continuous integrals in quantum field theory and statistical mechanics. - M .: Atomizdat , 1976 .-- 256 p.
Slavnov A. A., Faddeev L. D. Introduction to the quantum theory of gauge fields. - M .: Nauka , 1988 .-- 272 p.
Smolyanov O. G., Shavgulidze E. T. Continuous integrals. - M .: Nauka , 1990 .-- 150 p.
Feynman R., Hibs A. Quantum mechanics and path integrals. - M .: Mir , 1968 .-- 384 p.
Shestakova T.P. Continuous integral method in quantum field theory. - Izhevsk: IKI, 2005 .-- 228 p.

[1] The most typical example of an integration domain in a function space is the set of all functions of a given space that satisfy the condition of fixing their values at two points (at the ends of the segment).

[2] Article in the Physical Encyclopedia (A. A. Slavnov).

[_f65bb777a930da55-3] Polyakov, 1999 .