Linked Questions

13
votes
3answers
3k views

Eigenstate of field operator in QFT

Why don't people discuss the eigenstate of the field operator? For example, the real scalar field the field operator is Hermitian, so its eigenstate is an observable quantity.
10
votes
2answers
2k views

Derivative interaction: $\mathcal{H}_\mathrm{int}\neq - \mathcal{L}_\mathrm{int}$. Question about Feynman Rules

As we known, if there is time derivative interaction in $\mathcal L_\mathrm{int}$, then $\mathcal{H}_\mathrm{int}\neq -\mathcal{L}_\mathrm{int}$. For example, Scalar QED, $$ \begin{aligned} \mathcal{...
14
votes
1answer
898 views

Divergent integrals in QFT

I am starting to learn about QFT and something that I noticed is that integrals who would diverge otherwise are assigned a value if we do it by contour integration using the residues theorem and the ...
13
votes
1answer
1k views

What's the relation between path integral and Dyson series?

If one solves the Schrodinger equation $$i\hbar\partial_tU(t,0) = H U(t,0)$$ for time evolution operator $U(t,0)$, one can get the following Dyson series $$U(t,0) = \sum_n(\dfrac{-i}{\hbar})^n\...
7
votes
1answer
1k views

How to derive completeness relation in quantum field theory with a Lorentz invariant measure?

$\bullet$ 1. For the one-particle states, the completeness relation is given in Peskin and Schroeder, $$(\mathbb{1})_{1-particle}=\int\frac{d^3\textbf{p}}{(2\pi)^{3}}|\textbf{p}\rangle\frac{1}{2E_\...
6
votes
2answers
325 views

Definition of Sigma Model Path Integral

All references I have consulted have been extremely sketchy about this point. The (2 dimensional) nonlinear sigma model in some Riemannian manifold $M$ with metric $g_{\mu\nu}$ has action $$S = \frac{...
2
votes
2answers
262 views

Real scalar field and its quantum state: why are the diagonal components static here?

Consider a (free and massless) real scalar field $\phi(x)$ with Hamiltonian $$ H := \int d^3\mathbf{x}\; \bigg[ \frac{1}{2} \pi^2(\mathbf{x}) + |\nabla\phi(\mathbf{x})|^2 \bigg] $$ where $\pi(\mathbf{...
5
votes
1answer
382 views

Why the vacuum expectation value?

I am reading "QFT in a Nutshell", and the beginning of the book progresses like this: Show how $\langle q_F|e^{-iHt}|q_I\rangle=\int Dq\ e^{iS}$ Says that we are more interested in $\langle F|e^{-iHt}...
7
votes
1answer
175 views

Positive Definiteness of Killing Form in Gauge Theory

This question is related to requirement that the gauge group of a gauge theory be a direct product of compact simple groups and $U(1)$ factors but is not the same as, for example, this question (...
2
votes
0answers
50 views

What does the identity operator look like in QFT, written in momentum eigenstates? [duplicate]

This is a followup to a question I read recently: What does the identity operator look like in Quantum Field Theory? Out of curiosity, I was writing down what I figured to be the momentum basis ...