Feynman-Kac Formula: Exploring the Logic and Connection to the Path Integral

[PGaccount]

Take a path in spacetime. The amplitude for this path is e^iS. The paths which are very close to the classical path are the ones that contribute to the path integral. All these paths have approximately the same amplitude e^iS_classical. This means that the final amplitude is equal to Ne^iS_classical where N is the "number of paths" close to the classical one which constructively contribute to the path integral. This number is known to be G(0, 0), so we have

path integral = G(0, 0) e^iS_classical

This is called the semi-classical approximation. As a function of the time β between the endpoints, G(0, 0) ~ β^-1/2. Similarly, we have p_i = 1/Z e^-βE_i. 1/Z can be thought of as a multiplicity, and the probability of i is the product of this multiplicity and the probability factor e^-βE_i. The analog of this equation in terms of the density operator is ρ = 1/Z e^-βH Since Z = e^W[j], we have the expression e^-W[j]e^-βH for the density operator. The number density of particles per unit volume is 2E. This becomes 1/2E in momentum space, so there are dp/2E states in range dp of momenta. Instead of talking about paths, we can say that the particle "propagates from the initial to the final point with momentum p", and the corresponding amplitude would be e^ipxe^-iEt. This would have a multiplicity dp/2E in the path integral

∫ dp/2E e^ipxe^-iEt

which is a sum over all possible momenta with which the particle can propagate. In the path integral picture, the correlator of n operators is

$$ \langle O_1, ..., O_n \rangle = \frac{\int_{map(\Sigma, R)} O_1...O_n e^{-S}}{\int_{map(\Sigma, R)} e^{-S} } $$

The corresponding expression in the hamiltonian picture is the following, which is called the feynman-kac formula. The denominators in these two formulas are like Z in 1/Z e^-βH.

$$ \langle \phi(x_1), ..., \phi(x_n) \rangle = \frac{\mathrm{tr} \ e^{-x^0 H}\phi(x_1) e^{(x^0_1 - x^0_2)H} \phi(x_2)...\phi(x_n) e^{(x^0_n - \beta)H}}{\mathrm{tr} e^{-\beta H}}$$

what is the logic behind this formula, and how is it connected to the path integral version?

 

[eljose79]

The Feynman-Kac formula is a mathematical expression that relates the path integral formalism of quantum mechanics to the Hamiltonian formalism. It is essentially a way to calculate the expectation value of a product of operators in terms of the trace of the density operator.

The logic behind this formula is based on the idea that the path integral and the Hamiltonian formalism are two different mathematical representations of the same physical system. In the path integral, the system is described in terms of all possible paths that the system can take, while in the Hamiltonian formalism, the system is described in terms of the energy levels and the operators that act on them.

The connection between the two formalisms is made through the use of the density operator, which is a mathematical object that contains all the information about the system. In the path integral, the density operator is represented as a sum over all possible paths, while in the Hamiltonian formalism, it is represented as a trace over the energy levels.

The Feynman-Kac formula essentially combines these two representations by expressing the expectation value of a product of operators in terms of the density operator. This allows us to use the information from the path integral, such as the amplitude for a particular path, to calculate the expectation value in the Hamiltonian formalism.

In summary, the Feynman-Kac formula is a mathematical tool that connects the path integral and Hamiltonian formalism, allowing us to use the information from both representations to calculate expectation values.