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 $\times$, $+$!

522 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
14
votes
1answer
1k views

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}^\...
10
votes
0answers
539 views

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 ...
6
votes
0answers
105 views

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 $B$....
6
votes
0answers
299 views

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 ...
6
votes
0answers
136 views

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 ...
6
votes
2answers
291 views

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 (...
6
votes
0answers
168 views

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 $H$. ...
6
votes
0answers
343 views

Normal ordering and path integrals

What is the manifestation of normal ordering for creation/annihilation operators in the path-integral formalism?
5
votes
0answers
69 views

How to define an observable on a subsystem?

Suppose we have a bipartite system $AB$. I would intuitively say that an observable $O$ "acts as the identity on $A$" if $$ O=\mathbb 1_A \otimes O_B$$ But then I would also say this if $$ \...
5
votes
1answer
234 views

Quantization of Klein-Gordon field (what is creation operator there and what annihilation)

Recently in my class we studied quantization of fields and I'm brooding over an argument/ motivation on the construction of the quantization of the Klein-Gordon field. Recall the "classical" ...
5
votes
0answers
74 views

Convergence radius of the Operator Product Expansion in non-conformal QFTs

As by the Wikipedia definition, an OPE in QFT is a convergent expansion of the product of two operators at different spacetime points in terms of a sum of (possibly infinitely many) local operators. ...
5
votes
0answers
60 views

Projection operator (relative angular momentum) in FQHE Toy hamiltonian

I am working on Fractional Quantum Hall Effect and reading these lecture notes http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf. As all others sources I have found, none of them precisely define the ...
5
votes
1answer
226 views

Why is time-evolution unitary - The Heisenberg-picture Version

There are various versions of this question already on this site, which attempt to justify / make plausible that the time evolution of quantum mechanical observables is unitary. Most of these ...
5
votes
1answer
233 views

Why are all transformations of quantum operators inner automorphisms?

Operators in quantum mechanics are basically related to each other through their Lie algebra i.e. the commutator $\times \frac{1}{i\hbar}$. This is then connected to the state space i.e. the Hilbert ...
5
votes
0answers
503 views

Normal ordering in path integral of QFT

In QFT, we use normal ordering to eliminate infinity from hamiltonian. In path integral formulation of QFT though, since what we integrate over is "classical field configuration", instead of operators,...
5
votes
0answers
497 views

Action of Parity operator on Impulse representation

Is my derivation of the action of the parity operator $\mathbb{P}$ on the $|p\rangle$ representation correct? $$\left( \mathbb{P}\tilde\psi \right)(p)= - \tilde\psi (p).$$ Obtained from $$\left( \...
4
votes
1answer
156 views

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 ...
4
votes
0answers
140 views

Conditions for the Hamiltonian's spectrum to be discrete

I came across this article [1], 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,...
4
votes
1answer
58 views

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 ...
4
votes
0answers
114 views

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 ...
4
votes
1answer
132 views

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 $|\phi(x)|^2$ ...
4
votes
0answers
85 views

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#...
4
votes
0answers
55 views

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 ...
4
votes
0answers
327 views

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 ...
4
votes
0answers
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 ...
4
votes
0answers
179 views

Is time ordering defined for a single operator depending of two time variables?

The time ordering for the purpose of quantum mechanics is e.g. given by $${\mathcal T} \left[A(x) B(y)\right] := \begin{matrix} A(x) B(y) & \textrm{ if } & x_0 > y_0 \\ \pm B(y)A(x) & \...
4
votes
0answers
190 views

Non-Hermiticity when Fourier transforming onto a finite lattice

I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to ...
3
votes
0answers
53 views

General and geometric prescription of Picture-changing operator (PCO), Polchinski Vol.2, section 12.5

In section 12.5, Polchinski tried to give a general description of PCO from a super-riemann surface view. He gave the generalized amplitude, The measure on supermoduli space The expression $(5.4.19)$ ...
3
votes
0answers
82 views

Free field realisation for Vertex Operators

I wanted to know if for a generic CFT (in weak coupling limit) there exists free field realisation of the vertex operators. This seems like it should exist but I want to know a generic algorithm to ...
3
votes
0answers
118 views

Particle in a box with absolutely continuous spectrum

Let's consider a particle on a real line in a potential $V(x)$ which disappears at infinity. The Hamiltonian is: $$ H: W^{2,2}(\mathbb{R}) \subset L^2(\mathbb{R}) \to L^2(\mathbb{R}) \\ \big( H \, ...
3
votes
0answers
72 views

Vacuum state after diagonalising a free Hamiltonian?

The (free) hoping Hamiltonian $$ H = \alpha \sum_i^N c_i^{\dagger}c_i + \sum_{ij}^NV_{ij} c^{\dagger}_ic_j $$ is easily diagonalised by defining a new set of fermionic operators $$ d_a \doteq \sum_i^N ...
3
votes
1answer
113 views

How do *-Algebras correspond to operators on a Hilbert space?

In algebraic quantum field theory, a theory is defined through a net of observables $\mathcal{O} \mapsto \mathcal{A}(\mathcal{O})$ fulfilling the Haag-Kastler axioms (see e.g. this introduction, sec. ...
3
votes
0answers
61 views

Background on the Stone-von Neumann theorem

I'm actually a mathematician. I'm required to give a lecture on the Stone-von Neumann theorem. I already have all the mathematical details figured out, but I wish to make the lecture more interesting ...
3
votes
0answers
58 views

How do ladder operators in harmonic oscillator problem manage to accomplish this?

When we try to solve the harmonic oscillator problem by projecting the time independent equation onto position basis,we obtain solutions which do not vanish at infinity, then we ignore these solutions ...
3
votes
0answers
92 views

BRST invariant vertex operator

I'm trying to compute the commutator $\left[Q_{BRST}(z), V^{-1/2}_{v}(w)\right]$, where $V^{-1/2}_{v}(w)$ is the vertex operator corresponding to a massive fermion state. The vertex reads $$ V_{v}^{-\...
3
votes
0answers
54 views

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 ...
3
votes
0answers
79 views

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 ...
3
votes
0answers
52 views

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 ...
3
votes
1answer
245 views

Representing a rotation around an arbitrary axis using Wigner $D$-matrix

It is known that an arbitrary rotation can be expressed in terms of three consecutive rotations called the Euler rotations. So instead of expressing the rotation operator as $\hat{R}(\hat{n},\phi) = \...
3
votes
0answers
67 views

What Operator is the Superconformal Index Counting?

Given a differential operator $\mathcal{D}$ with adjoint $\mathcal{D}^\dagger$, the analytical index of $\mathcal{D}$ is usually defined by $$\text{ind }\mathcal{D}=\dim\ker\mathcal{D}-\dim\ker\...
3
votes
0answers
95 views

Inverse of time-ordered exponential

It is straightforward to show that $$ \left[T_\leftarrow\exp\left(-i\int_0^tH(\tau)d\tau\right)\right]^{-1} = T_\rightarrow\exp\left(i\int_0^tH(\tau)d\tau\right), $$ where $T_\leftarrow$ and $T_\...
3
votes
1answer
171 views

When does a Hermitian operator have real matrix elements?

I will use braket-notation, but my question is not specific to quantum mechanics. Instead, I would be interested in a general answer for operators in some Hilbert space. Let $H$ be a Hermitian ...
3
votes
0answers
86 views

State-operator correspondence in 2D CFT

In CFT we can express a state in terms of it's operator by $$ |\phi\rangle=\lim_{z,\bar{z}\rightarrow 0} \phi(z,\bar{z}) |0\rangle $$ Now in 2D we can expand our field in terms of $\phi_n$ as $$ \phi(...
3
votes
0answers
64 views

Convert a Lindbladian time evolution operator to the Kraus operator sum representation

I try to understand how I can convert a Lindbladian time evolution operator to the corresponding Kraus operator sum. Let's assume we have a time independent Hamiltonian $H$ and a set of time ...
3
votes
0answers
65 views

Experimental tests of operator ordering

In quantization, we frequently run into ordering ambiguities. In general, this means that there can be inequivalent quantum theories corresponding to the same classical theory. Has there ever been an ...
3
votes
1answer
112 views

In General Relativity, can I represent a Tetrad/Frame field in terms of ladder operators?

I've been interested in expressing the metric tensor $g$ in terms of it's harmonic expansions. In particular I'm interested in writing the tetrad/frame-fields in terms of such expansions. For ...
3
votes
1answer
334 views

How to find ladder operators that diagonalize a Hamiltonian in QFT?

I have some trouble understanding how one can, in the context of QFT, diagonalize a Hamiltonian $H$ by the introduction of ladder operators $a$ and $a^\dagger$ (I have trouble understanding how one is ...
3
votes
0answers
112 views

Interpretation of annihilation and creation operators

If we write some quantum field in a form using creation and annihilation operators we are, in a way, doing a Fourier series with annihilation and creation operators being coefficients. So, if they are ...
3
votes
0answers
72 views

Given two abitrary operators satisfying the canonical commutation relations at all times, can I show that they obey the same time evolution?

To be more clear: I take two operators, $A$ and $B$, which are functions of time, $A(t)$ and $B(t)$. I assume their time development to be unitary (but not neccessarily the same): $A$'s time evolution ...
3
votes
0answers
159 views

Fermion creation operator in boson basis

I've been reading Giamarchi, Quantum Physics in One Dimension, Chapter 2 on 1d bosonization, and in appendix B.1, he derives equation B.2, which represents the fermion creation operator $\psi_r (x)$ ...

1
2 3 4 5
11