5
votes
2answers
123 views

Eigenstates of a Hermitian field operator

Consider a Hermitian field operator $\phi(x)$ with eigenstates satisfying $$ \phi(x) |\alpha\rangle = \alpha(x) | \alpha \rangle $$ I'm trying to determine the inner product between the eigenstates. ...
2
votes
1answer
118 views

Where does this delta of zero come from?

It is common when evaluating the partition function for a $O(N)$ non-linear sigma model to enforce the confinement to the $N$-sphere with a delta functional, so that $$ Z ~=~ \int d[\pi] d[\sigma] ~ ...
0
votes
0answers
111 views

Derivation of (2.45) in Peskin and Schroeder

I'm having trouble understanding the step $$\left[\pi (\vec{x},t),\int d^{3}y ~(\frac{1}{2} \pi (\vec{y},t)^{2}+\frac{1}{2}\phi (\vec{y},t)(-\nabla^{2} +m^{2})\phi (\vec{y},t)) \right]$$ $$ =\int ...
3
votes
2answers
71 views

Application of Poisson summation Formula to Casimir Effect

I'm studying the Casimir Effect at finite temperature. To calculate the Helmoltz free energy in the canonical ensemble I need to sum a particular series. In some scientific papers it is suggested to ...
4
votes
2answers
106 views

Picture of supports

This questions stems from Axiomatic Quantum Field Theory and is mathematical in nature. However, I feel that an answer from physicists is more in line with what I will be asking. Let $\phi$ be a ...
3
votes
3answers
345 views

Multivariable Dirac Delta and Faddeev-Popov Determinant

From this mathstack page and in particular Qmechanic's answer: There exists an $n$-dimensional generalization $$\tag{1} \delta^n({\bf f}({\bf x})) ~=~\sum_{{\bf x}_{(0)}}^{{\bf f}({\bf ...
3
votes
1answer
198 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 ...
2
votes
1answer
155 views

Justification for smeared fields in the Wightman axioms?

I just started reading PCT, Spin and Statistics, and All That. Can someone explain why we use operator valued distributions to describe fields? I read somewhere that it would take infinite energy to ...
2
votes
1answer
209 views

Getting rid of double delta function in Feynman rules

[1] A very simple example of feynman rule for scalar fields. After computing the diagram i have got the following: $$ -i(2\pi)^4g^2\int d^4q \frac{i}{q^2 -m^2c^2}\delta^{(4)}(p_1 - p_3 -q) ...
6
votes
1answer
274 views

Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

Are the following operations O.K.? This is related to the Feynman parameter trick. $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using ...
3
votes
1answer
331 views

Crazy Dirac Deltas

I'm not expecting any rigor in the following and the answers...since we're dealing with Dirac deltas in the context of QFT. Consider the integral $$ \int d^4q\ \Theta(q_0)\Theta(p_{3,0}+q_0)\ ...