# Questions tagged [operators]

In physics, an operator is almost always either a square matrix or a linear mapping between two function spaces (defined on, say, $\mathbb R^n$). Operators serve as observables and as time evolution operators in Quantum Mechanics. This tag will most often find valid use in quantum mechanics; don't use this tag just because your equations contain "everyday operations" like $\times$, $+$!

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### How can I write a Gaussian state as a squeezed, displaced thermal state?

I would like to write a Gaussian state with density matrix $\rho$ (single mode) as a squeezed, displaced thermal state: \begin{gather} \rho = \hat{S}(\zeta) \hat{D}(\alpha) \rho_{\bar{n}} \hat{D}^\...
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### Wick theorem and OPE

I'm trying to work out in detail how the Wick theorem is used for constructing OPEs in CFT. One of the first things which bothers me is the difference in definitions of normal ordered product and ...
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### Baryonic operators in ${\cal N}=1$ $U(N)$ SQCD in four dimensions

Seiberg's duality is usually considered as a duality for $SU(N_c)$ theories with $N_f$ flavors. In his case, the vacuum for $N_f \geq N_c$ is parameterized by mesons $M$ and baryons ${\bar B}$ and $B$....
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### Is there a physical significance to non-normal states of the algebra of observables?

Quantum theory may be formalized in several different ways. Generally, the physical discussion of different states of a quantum system distinguishes pure and mixed states, and then subsumes both in a ...
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### What quantum measurement formalism is easiest to implement physically?

As part of my studies and research, I have learned to work with three different measurement formalism which I define to avoid any ambiguity with the nomenclature: General measurements, which are ...
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### Where can I find a detailed derivation of the form of two-body operators in the second quantization?

I've been looking around online for a couple hours now and I can't find a very informative derivation of the form for two body operators in the second quantization. Is there a resource online (...
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### What is the physical interpretation of the automorphism on bounded operators induced by an S matrix?

In a QFT, the S-matrix $S$ is a unitary operator, that fixes the vacuum and commutes with the unitary operators implementing the action of the Poincare group on an appropriate Hilbert space $H$. ...
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### Normal ordering and path integrals

What is the manifestation of normal ordering for creation/annihilation operators in the path-integral formalism?
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### What is the set of observables of a quantum system?

This is a question I am wondering about because the answer to it seems to have some interesting - but perhaps already long considered and dismissed because it's been settled - implications for the ...
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### Conditions for the Hamiltonian's spectrum to be discrete

I came across this article , in which the author studies some Hamiltonian that have a discrete spectrum even though they do not go to infinity at infinity. In there, the author makes several claims,...
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### Boundary conditions, physical wave-functions and domain of Hamiltonian

Context : In quantum mechanics, time evolution is described by a one-parameter unitary group, acting on the Hilbert space of states. Under Stone's theorem (with the right hypotheses), this group has a ...
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### What does it mean for an extended operator to possess "local excitations"?

In the context of defect conformal field theory, we consider in operator product expansions "local excitations" of the defect (see e.g. text between eq. $(1.1)$ and $(1.2)$ in the paper ...
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### Making connection of first quantization to second quantization, (quantum field theory)

In first quantization, a state of system is represented by wavefunction (w.f.) $\phi(x)$ (a representation of a state $|\phi\rangle$ in Hilbert space). The way I understand it is that $|\phi(x)|^2$ ...
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### Why is the generalized momentum replaced by the momentum operator but not the ordinary momentum?

I was trying to understand the derivation of the Hamiltonian for a charged particle in an electromagnetic field. https://en.wikipedia.org/wiki/Hamiltonian_mechanics#...
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### What is a singular continuous spectrum?

I read some answers about this and the wikipedia page that basically always say that a spectrum can be decomposed into: $$\mu = \mu_{ac} + \mu_{sc} + \mu_{pp},$$ where $\mu_{ac}$ is absolutely ...
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### Completeness relation and scalar product for Grassmann coherent states

I was wondering, given two fermionic canonical operators: $$\{\hat{\chi}_{\alpha a}(x),\hat{\bar{\chi}}_{\beta b}(y)\} = \delta_{\alpha\beta}\delta_{ab}\delta^{(3)}(x-y)$$ they should have as ...
247 views

### Nature of Microscopic space-time

I am going through the introductory chapter's of Schwinger's Source theory. He writes, It [Source Theory] is a phenomenological theory, designed to describe the observed particles. No speculations ...
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### Issues of infinity with time ordering of the interaction Hamiltonian of $\phi^4$ theory

I use $\phi^4$ theory as an example here, but similar things happen to other theories as well. In $\phi^4$ theory, we can easily use the Lagrangian here to write down the interaction Hamiltonian (in ...
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### Partition function $\mathrm{tr} e^{-\beta H}$ for non-trace-class operator

For Hamiltonian $H$, the partition function is defined as $$Z=\mathrm{tr} e^{-\beta H}.$$ However, consider a free particle $H = -\Delta$ in $\mathbb R^1$. In this case, the operator $e^{-\beta H}$ is ...
### Convergence of $c^* Ac$ in second quantization
Let $A$ denote a bounded operator on a complex separable Hilbert space $\mathscr{H}$. Let $\mathscr{F} = \bigoplus \mathscr{F}_N$ be the Fock space generated by $\mathscr{H}$ where $\mathscr{F}_N$ is ...