Integral of way

An integral of way (“ path integral English ”) is a functional Intégrale , i.e. integrating it is a functional calculus and that the sum is taken on functions, and not on real numbers (or complexes) as for the ordinary integrals. There is thus here business with an integral in infinite dimension. Thus, one will carefully distinguish the integral from way (integral functional calculus) of an ordinary integral calculated on a way of physical space, that the mathematicians call curvilinear Intégrale.

It is Richard Feynman which introduced the integrals of way in physics into its thesis, supported in May 1942, bearing on the formulation of the quantum Mécanique based on the Lagrangien. Because of the Second world war, these results will be published only in 1948. This mathematical tool quickly was essential in theoretical physics with its generalization on the Quantum theory of the fields, in particular allowing a quantification of the not-abelian theories of gauge simpler than the canonical procedure of quantification.

In addition, the mathematician Mark Kac then developed a similar concept for the theoretical description of the Brownian Movement, taking as a starting point results obtained by Norbert Wiener in the years 1920. One speaks in this case about the Formule about Feynman-Kac, which is an integral for the measurement of Wiener.

Genesis of the concept of integral of way

Whereas it is student of 3rd cycle under the direction of Wheeler at the university of Princeton, the young person Feynman seeks a method of quantification based on the Lagrangien to be able to describe a system not having necessarily a Hamiltonien. Its motivation first is to quantify the new formulation of the traditional electrodynamics based on the remote action which it has just developed with Wheeler.

In spring of 1941, it meets Herbert Jehle, then visitor with Princeton, which indicates to him at the time of one evening to the Nassau Tavern the existence of an article of Dirac which precisely discusses the quantification from the Lagrangian one. Precise Jehle with Feynman which this formulation allows a relativistic approach covariante much easier than that based on the Hamiltonian. The following day, the two physicists go to the library to study the article of Dirac. They read there in particular the following sentence: for two moments $t\; \backslash ,$ and $t\; +\; \backslash \; close\; epsilon\; \backslash ,$, the elementary amplitude of transition:

In this formula, the size $S\; \backslash ,$ is the traditional action:

In order to include/understand what Dirac want to say by similar , Feynman studies the case of a nonrelativistic particle of mass $m\; \backslash ,$ for which the Lagrangian one is written:

It is known that:

Feynman supposes then a relation of proportionality :

where $A\; \backslash ,$ is a constant unknown factor. In the presence of Jehle, Feynman shows that this equation implies that $\backslash \; psi\; (Q,\; T)\; \backslash ,$ obeys the equation of Schrödinger:

in the condition which the constant unknown factor $A$ is equal to:

With the autumn 1946, at the time of the bicentenary of the university of Princeton, Feynman met Dirac the ascetic and the following exchange took place between the two geniuses:

- Feynman: “Did you Know that these two sizes were proportional? ”

- Dirac: “They are it? ”

- Feynman: “Yes. ”

- Dirac: “Oh! It is interesting. ”

This laconic answer will put a final point at the discussion… For more historical details, one will read with profit the article of Schweber.

Recalls on the propagator of the equation of Schrödinger

To simplify the notations, one restricts oneself below with the case of only one dimension of space. The results extend without difficulty with an unspecified number of dimensions.

Definition

Let us consider a particle of mass $m\; \backslash ,$ not relativist, described in quantum Mécanique by a Fonction of wave. Let us suppose that one gives oneself the initial condition $\backslash \; psi\; (q\_0,\; t\_0)\; \backslash ,$ at one initial moment $t\_0\; \backslash ,$ fixed. Then, the function of wave at one later moment $t\_1\; >\; t\_0\; \backslash ,$, solution of the equation of Schrödinger, is given by the integral equation:

where $K\; (q\_1,\; t\_1|q\_0,\; t\_0)\; \backslash ,$ is the propagating of the particle:

$\backslash \; hat\; \{H\}$ is the Hamiltonian operator of the particle.

Equation of Chapman-Kolmogorov

Let us recall that, if $t\_2\; >\; t\_1\; >\; t\_0\; \backslash ,$, the propagator obeys the equation of Chapman-Kolmogorov:

This relation will enable us to find the expression of the propagator in term of an integral of way.

Expression of the propagator in term of integral of way

Let us seek the expression of the propagator $K\; (q\_f,\; t\_f|q\_i,\; t\_i)\; \backslash ,$ between the initial moment $t\_i\; \backslash ,$ and the final moment $t\_f\; \backslash ,$.

Application of the equation of Chapman-Kolmogorov

One cuts out the time interval $\backslash \; Delta\; T\; =\; t\_f\; -\; t\_i\; \backslash ,$ in $N\; \backslash ,$ elementary time intervals of duration $\backslash \; epsilon\; \backslash ,$ by introducing the $N\; +\; 1\; \backslash ,$ urgent:

with $t\_0\; =\; t\_i\; \backslash ,$ and $t\_N\; =\; t\_f\; \backslash ,$. There is thus $N\; -\; 1\; \backslash ,$ urgent intermediaries $t\_k\; \backslash ,$ between the initial moment $t\_0\; \backslash ,$ and the final moment $t\_N\; \backslash ,$. So that the time intervals have one duration $\backslash \; epsilon\; =\; t\_\; \{k+1\}\; -\; t\_k\; \backslash ,$ elementary, the limit $N\; \backslash \; to\; +\; \backslash \; infty\; \backslash ,$ is implied.

The application of the equation of Chapman-Kolmogorov first once makes it possible to write:

then, by applying it second once:

and so on. One obtains with final after $N\; -\; 1\; \backslash ,$ applications to the $N\; -\; 1\; \backslash ,$ intermediate times:

One is thus brought to consider the elementary propagating :

Elementary propagator: formulate of Feynman-Dirac

For a particle of mass $m\; \backslash ,$ not relativist with a dimension in a potential, whose Hamiltonien operator is written:

and the elementary propagator is written:

One uses the Formule of Trotter-Kato:

This formula is not commonplace, because the operators $\backslash \; hat\; \{has\}\; \backslash ,$ and $\backslash \; hat\; \{B\}\; \backslash ,$ does not commutate in general! One obtains here:

One can leave the exponential one containing the potential which depends only on the position:

The remaining element of matrix is the propagator of the free particle , therefore one can finally write the expression:

However the expression of the free propagating is known exactly:

It is noticed that the argument of exponential can be rewritten in term of a discretized expression of the speed :

like:

One from of deduced that the elementary propagator is written:

The arguments of the two exponential ones being now complex numbers, one can write without difficulties:

that is to say still:

The term between bracket represents Lagrangian particle:

from where the formula of Feynman-Dirac for the elementary propagator:

Integral of way

One injects the expression of Feynman-Dirac in the general formula:

It comes:

The argument of exponential being complex numbers, one can write:

One recognizes in the argument of exponential discretization of the traditional action :

One from of deduced with Feynman the expression from the propagator like functional integral on all the continuous ways:

with formal measurement:

Interpretation

The formula of Feynman:

admits following interpretation: to calculate the amplitude of transition from the initial point $q\_i\; \backslash ,$ at the moment $t\_i\; \backslash ,$ towards the final point $q\_f\; \backslash ,$ at the moment $t\_f$, it is necessary to consider all the continuous ways $q\; (T)\; \backslash ,$ checking the boundary conditions: $q\; (t\_i)\; =\; q\_i\; \backslash ,$ and $q\; (t\_f)\; =\; q\_f\; \backslash ,$. Each way is seen allotting a complex “weight” of module unit: $\backslash \; exp\; (I\; S\; \backslash \; hbar)\; \backslash ,$, where $S\; \backslash ,$ is the traditional action calculated on this way. One “then summons” this noncountable infinity of complex weights, and one obtains in fine the amplitude of desired transition.

This interpretation is the work of Feynman alone, Dirac not having crossed the step. It is implicit in its thesis of 1942, and explicit in the publication of 1948.

Semi-traditional limit

Within the limit where the action of the system is much larger than $\backslash \; hbar$, one can use a development of the traditional semi type, where $y$ is a small disturbance of the traditional trajectory $x\_\; \{C\}$: $x=x\_\; \{C\}\; +y$

Let us consider a Lagrangian standard:

$\backslash \; mathcal\; \{L\}\; =\; \backslash \; frac\; \{m\; \backslash \; dowry\; \{X\}\; ^\; \{2\}\}\; \{2\}\; -\; V\; (X)$

One then writes the action in the following form, while limiting oneself to the second order:

$S\; \backslash \; approx\; S+\; \backslash \; int\_\; \{t\_\; \{I\}\}\; ^\; \{t\_\; \{F\}\}\; dt\; \backslash \; underbrace\; \{\backslash \; frac\; \{\backslash \; delta\; S\}\; \{\backslash \; delta\; X\; (T)\}|\_\; \{x\_\; \{C\}\}\}\; \_\; \{=0\}\; there\; (T)\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; int\_\; \{t\_\; \{I\}\}\; ^\; \{t\_\; \{F\}\}\; dt\_\; \{1\}\; dt\_\; \{2\}\; \backslash \; frac\; \{\backslash \; delta^\; \{2\}\; S\}\; \{\backslash \; delta\; X\; (t\_\; \{1\})\; \backslash \; delta\; X\; (t\_\; \{2\})\}|\_\; \{x\_\; \{C\}\}\; there\; (t\_\; \{1\})\; there\; (t\_\; \{2\})\; \backslash \; Longrightarrow$

$S\; \backslash \; approx\; S+\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; int\_\; \{t\_\; \{I\}\}\; ^\; \{t\_\; \{F\}\}\; dt\; (m\; \backslash \; dowry\; \{there\}\; ^\; \{2\}\; -\; V$ (x_ {C}) y^ {2})

one can thus approximate the propagator:

$K\; (x\_\; \{F\},\; t\_\; \{F\};\; x\_\; \{I\},\; t\_\; \{I\})\; \backslash \; approx\; e^\; \{I\; S\; \backslash \; hbar\}\; \backslash \; int\; \backslash \; mathcal\; \{D\}\; e^\; \{I\; \backslash \; int\_\; \{t\_\; \{I\}\}\; ^\; \{t\_\; \{F\}\}\; dt\; (m\; \backslash \; dowry\; \{there\}\; ^\; \{2\}\; -\; V$ (x_ {C}) y^ {2}) /2 \ hbar}

an integration by part of the exhibitor brings back to a Gaussienne form:

$K\; (x\_\; \{F\},\; t\_\; \{F\};\; x\_\; \{I\},\; t\_\; \{I\})\; \backslash \; approx\; e^\; \{I\; S\; \backslash \; hbar\}\; \backslash \; int\; \backslash \; mathcal\; \{D\}\; e^\; \{I\; \backslash \; int\_\; \{t\_\; \{I\}\}\; ^\; \{t\_\; \{F\}\}\; dt\; (there\; \backslash \; frac\; \{d^\; \{2\}\}\; \{dt^\; \{2\}\}\; -\; V\; \text{'}\text{'}\; (x\_\; \{C\}\; there)\; /2\; \backslash \; hbar\}$

Let us define the operator $\backslash \; hat\; \{O\}\; =\; -\; m\; \backslash \; frac\; \{d^\; \{2\}\}\; \{dt^\; \{2\}\}\; -\; V$ (x_ {C})

the rules of calculation of the Gaussiennes integrals provide:

$K\; (x\_\; \{F\},\; t\_\; \{F\};\; x\_\; \{I\},\; t\_\; \{I\})\; \backslash \; approx\; cste\; \backslash \; cdot\; \backslash \; sqrt\; \{\backslash \; frac\; \{1\}\; \{Det\; (\backslash \; hat\; \{O\})\}\}\; \backslash \; cdot\; e^\; \{I\; S\; \backslash \; hbar\}$

Now let us consider the function $\backslash \; Psi\; (T)$ definite as follows:

$\backslash \; hat\; \{O\}\; \backslash \; Psi=\; (-\; m\; \backslash \; frac\; \{d^\; \{2\}\}\; \{dt^\; \{2\}\}\; -\; V$ (x_ {C})) \ Psi=0

with the conditions of edges:

$\backslash \; Psi\; (t\_\; \{I\})\; =0$

$\backslash \; Psi\text{'}\; (t\_\; \{I\})\; =1$

One can then show that:

$Det\; (\backslash \; hat\; \{O\})\; =\; cste\; \backslash \; cdot\; \backslash \; Psi\; (t\_\; \{F\})$

what gives us for the approximation of the propagator:

$K\; (x\_\; \{F\},\; t\_\; \{F\};\; x\_\; \{I\},\; t\_\; \{I\})\; \backslash \; approx\; \backslash \; sqrt\; \{\backslash \; frac\; \{has\}\; \{\backslash \; Psi\; (t\_\; \{F\})\}\}\; \backslash \; cdot\; e^\; \{I\; S\; \backslash \; hbar\}$

one determines the constant one has starting from the propagator of the free particle:

$K\_\; \{FP\}\; (x\_\; \{F\},\; t\_\; \{F\};\; x\_\; \{I\},\; t\_\; \{I\})\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{m\}\; \{2\; \backslash \; pi\; I\; \backslash \; hbar\; (t\_\; \{F\}\; -\; t\_\; \{I\})\}\}\; e^\; \{I\; S\; \backslash \; hbar\}$

in the case of the free particle, the function $\backslash \; Psi$ which satisfies the exposed conditions higher is $\backslash \; Psi\; (T)\; =t-t\_\; \{I\}$, which immediately gives us an expression for A. One obtains finally the approximation known as semi-traditional of the propagator:

$K\; (x\_\; \{F\},\; t\_\; \{F\};\; x\_\; \{I\},\; t\_\; \{I\})\; \backslash \; approx\; \backslash \; sqrt\; \{\backslash \; frac\; \{m\}\; \{2\; \backslash \; pi\; I\; \backslash \; hbar\; \backslash \; Psi\; (t\_\; \{F\})\}\}\; \backslash \; cdot\; e^\; \{I\; S\; \backslash \; hbar\}$

this approximation is powerful, and can even give an exact result sometimes, as for example if the potential is that of a harmonic oscillator of frequency $\backslash \; Omega$. In this case, the function $\backslash \; Psi$ must satisfy, in addition to the conditions of edge:

$(-\; m\; \backslash \; frac\; \{d^\; \{2\}\}\; \{dt^\; \{2\}\}\; -\; m\; \backslash \; omega^\; \{2\})\; \backslash \; Psi=0\; \backslash \; Longrightarrow\; \backslash \; Psi\; (T)\; =\; \backslash \; frac\; \{\backslash \; sin\; \backslash \; Omega\; (t-t\_\; \{I\})\}\{\backslash \; Omega\}$

and one obtains the exact expression of the propagator of the harmonic oscillator, by the semi-traditional approximation:

$K\_\; \{Ho\}\; (x\_\; \{F\},\; t\_\; \{F\};\; x\_\; \{I\},\; t\_\; \{I\})\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{m\; \backslash \; Omega\}\; \{2\; \backslash \; pi\; I\; \backslash \; hbar\; \backslash \; sin\; \backslash \; Omega\; (t\_\; \{F\}\; -\; t\_\; \{I\})\}\}\; \backslash \; cdot\; e^\; \{I\; S\; \backslash \; hbar\}$

with the traditional action of the harmonic oscillator:

$S\_\; \{Cl\}\; =\; \backslash \; int\_\; \{t\_\; \{I\}\}\; ^\; \{t\_\; \{F\}\}\; dt\; \backslash \; mathcal\; \{L\}\; =\; \backslash \; frac\; \{m\; \backslash \; Omega\}\; \{2\}\; \backslash \; left\; (t\_\; \{F\}\; -\; t\_\; \{I\})\; -\; \backslash \; frac\; \{2x\_\; \{I\}\; x\_\; \{F\}\}\; \{\backslash \; sin\; \backslash \; Omega\; (t\_\; \{F\}\; -\; t\_\; \{I\})\}\backslash \; right$

to note another equivalent formulation of the approximation semi-traditional, known as of Van Vleck-Pauli-Morette, who rises directly from the preceding one:

$K\; (x\_\; \{F\},\; t\_\; \{F\};\; x\_\; \{I\},\; t\_\; \{I\})\; \_\; \{Sc\}\; =\; \backslash \; sqrt\; \{-\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\; I\; \backslash \; hbar\}\; \backslash \; frac\; \{\backslash \; partial\; ^\; \{2\}\; S\_\; \{partial\; Cl\}\}\; \{\backslash \; x\_\; \{I\}\; \backslash \; partial\; x\_\; \{F\}\}\}\; \backslash \; cdot\; e^\; \{I\; S\_\; \{Cl\}/\backslash \; hbar\}$

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