Questions tagged [functional-determinants]

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19
votes
4answers
4k views

Gelfand-Yaglom theorem for functional determinants

What is the 'Gelfand-Yaglom' Theorem? I have heard that it is used to calculate Functional determinants by solving an initial value problem of the form $Hy(x)-zy(x)=0$ with $y(0)=0$ and $y'(0)=1$. ...
5
votes
2answers
991 views

How to directly evaluate path integral for harmonic oscillator by brute force method?

It is easy to evaluate the green's function using path integral approach by evaluating classical action and using functional calculus method. Is it possible to evaluate path integral for harmonic ...
11
votes
3answers
751 views

How does the functional measure transform under a field redefinition?

My question is: how does the path integral functional measure transform under the following field redefinitions (where $c$ is an arbitrary constant and $\phi$ is a scalar field): \begin{align} \phi(x)&...
8
votes
1answer
2k views

Path integral as a functional determinant

In Peskin and Schroeder on pg. 304, the authors call the fermionic path integral: \begin{equation} \int {\cal D} \bar{\psi} {\cal D} \psi \exp \left[ i \int \,d^4x \bar{\psi} ( i \gamma_\mu D^\mu - m ...
8
votes
1answer
538 views

Lack of Maslov index in the path integral formalism

Introduction Consider Feynman's famous path integral formula \begin{equation} K(x_a,x_b) = \int \mathcal{D}[x(t)] \exp \left[ \frac{i}{\hbar} \int_{t_a}^{t_b} dt \, \mathcal{L}(x(t),\dot{x}(t),t) \...
2
votes
0answers
249 views

A question about Gel'fand-Yaglom method of calculating functional determinants

I know that the Gel'fand-Yaglom method is a way to calculate determinants of 1D differential operators. For instance, let us consider an operator $-\partial_r^2+W(r)$. Let us define $\psi_{0}(r)$ as ...
4
votes
1answer
353 views

Free space propagator: reconciling two results

In quantum mechanics, the free space propagator $G(q_f=0,q_i=0;\tau)$ can be easily calculated to be $$\sqrt{\frac{m}{2\pi i \hbar \tau}}$$ by inserting an identity operator. However if we use ...
3
votes
1answer
621 views

Determinant as a fermionic path integral

I understand that the determinant of a matrix can be written in terms of a fermionic path integral. The expression is: $$Z = \int D\bar{\psi}D\psi e^{-\iint d^4x' d^4x \bar{\psi}(x')B(x',x)\psi(x)}\...
1
vote
1answer
308 views

Vacuum Energy Calculation using Path Integral

I am currently reading Zee's book on quantum field theory, and I am in the chapter where he is introducing Grassmann integrals. He re-introduces the path integral evaluated for the vacuum, i.e. no ...