3
votes
1answer
120 views

Paths in the path integral

In the path integral approach one defines in some heuristic way the functional path integral \begin{equation} Z=\int{\cal{D}}\phi e^{iS(\phi)} \end{equation} and the one claims that one must ...
1
vote
0answers
69 views

From Minkowski to Euclidean Time in Path Integrals

I'm trying to prove the following equality: $$ <x_{f},\, it_{f}|x_{i},\, it_{i}>=\mathcal{N}\int_{\left\{ x\in\mathbb{R}^{\mathbb{R}}:\, x\left(t_{f}\right)=x_{f}\wedge ...
1
vote
0answers
117 views

Divergent path integral

What does it mean to have a divergent path integral in a QFT? More specifically, if $$\int e^{i S[\phi]/\hbar} D\phi (t)=\infty $$ What does this mean for the QFT of the field $\phi $? The field ...
2
votes
0answers
80 views

Does the success of canonical quantization guarantee the path integral works too?

If the canonical quantization approach to a field theory is successful, is it a good indicator that the path integral will work as well? Furthermore, can the success of a particular quantization ...
3
votes
1answer
264 views

Path Integral on a circle (calculation of phase and linear independance)

I am reading Schulman's "Techniques and applications of path integration" chapter on Path integrals on multiply-connected spaces. In the first section he calculates the path integral of a free ...
2
votes
1answer
104 views

Wick's theorem again

Could someone please elaborate on the accepted answer to this mathoverflow post? I'm working on a problem that looks like this \begin{equation}I=\int d^{n} x\, f(\vec x)\, e^{-\frac{1}{2} \vec x ...
4
votes
2answers
378 views

Normalization of the path integral

When one defines the path integral propagator, there is the need to normalize the propagator (since it would give you a probability density). There are two formulas which are used. 1) Original ...
2
votes
1answer
143 views

Delta functional in path integral

I've recently encountered a path integral of the form $$\int \delta[a\phi+b\phi']\,L(\phi,\phi')\;\mathcal D\phi\mathcal D\phi'$$ (where $a$, $b$ are integers) and would like to eliminate one of the ...
4
votes
1answer
253 views

Calculation of the spherical harmonic sum in the propagator of the particle on a sphere

I am calculating the propagator of the free particle on a sphere : $K(\theta_f \phi_f t_f; \theta_i \phi_i t_i)$. The wavefunctions in this case are the spherical harmonics $Y_{lm}(\theta, ...
3
votes
1answer
161 views

Diagonalizing/eigenvalues of the infinite dimensional matrix of N harmonic oscillators on a ring

I have trying to show that the continuum limit of N quantum harmonic oscillators gives rise the the klein-gordon field. However, instead of a usual finite string, I want to do it on a ring. Hence, my ...
2
votes
2answers
405 views

Intuition for Path Integrals and How to Evaluate Them

I'm just starting to come across path integrals in quantum field theory, and want to get the right intuition for the them from the start. The amplitude for propagation from $x_a$ to $x_b$ is typically ...
2
votes
1answer
315 views

Why can't the functional integral be derived in a mathematically rigorous way?

Why can't the functional integral be derived in a mathematically rigorous way? What are the obstacles that we have to overcome in order to achieve that goal?
8
votes
2answers
199 views

Are there rigorous constructions of the path integral for lattice QFT on an infinite lattice?

Lattice QFT on a finite lattice* is a completely well defined mathematical object. This is because the path integral is an ordinary finite-dimensional integral. However, if the lattice is infinite, ...
6
votes
2answers
369 views

Lagrangians combining terms with 1 and 2 derivatives

How are field theory Langrangians treated when some terms have 2 derivatives but others have only 1? Because the number of derivatives in a Lagrangian term is more easily even than odd, the ...