In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

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Role of physics in the zeta function $\zeta$ and the Riemann hypothesis

Hilbert and Polya suggested a physical way to verify the Riemann hypotesis about $\zeta(x)$. If the Riemann hypotesis is true, we can state all eigenvalues of physical problems are real. What is the ...
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89 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 $H$. ...
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127 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}} ...
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98 views

Derivation of the Lippmann-Schwinger equation

I was trying to understand the derivation of the Lippmann-Schwinger equation in Sakurai's Modern Quantum Mechanics, Section 6.1. Our teacher presented a much simpler derivation, similar to that on ...
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228 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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258 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( ...
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103 views

Density matrix formalism and group representation

The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space $\mathcal{H}$. Composition is defined ...
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137 views

Convention in physics for [],{} and operators (QM)

I got a little mixed up with the convention in physics. Usually a hat means an operator. For a given electron-ion Hamiltonian $\hat{H}_{e-n}$, what are the difference between these: 1) ...
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47 views

Self-adjoint extensions with 'teletransporting' boundary conditions

When choosing a self-adjoint extension of a Hamiltonian, in general one can obtain domains in which (i) the probabilities teleport* between points on the boundary and (ii) boundary conditions ...
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92 views

Hamiltonian Operator Interpretation of Quantum Anomaly

We can see the definition of quantum anomaly in terms of Lagrangian path integral formulation. What is the definition of quantum anomaly in terms of Hamiltonian operator approach or even more directly ...
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127 views

Virasoro Operators commutation relations

For the commutation relation in quantising the bosonic string $\left[L_n,L_{m}\right]=(n-m)L_{n+m}+\frac{D}{12}n(n^2-1)\delta_{n+m,0}$ we can then calculate this for $m=-n$ in between the vacuum ...
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83 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 ...
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82 views

Spectrum of a quantum relativistic “distance squared” operator

This question disusses the same concepts as that question (this time in quantum context). Consider a relativistic system in spacetime dimension $D$. Poincare symmetry yields the conserved charges $M$ ...
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65 views

How to do time evolution of operators in the Heisenberg Picture while staying in the Heisenberg Picture

Consider the time evolution of an operator in the Heisenberg picture: $$\tag{1}i\hbar \frac{d}{d t} \hat{A}_{H}(t) = \left([ \hat{A}_S(t), \hat H_S (t)] + i\hbar \frac{d}{d t} \hat{A}_S(t) ...
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91 views

Lorentz transformation - Bjorken & Drell

I'm trying to derive (14.25) in Bjorken & Drell (B&D) QFT. This is $$\tag{14.25}U(\epsilon)A^\mu(x)U^{-1}(\epsilon) = A^\mu(x') - \epsilon^{\mu\nu}A_\nu(x') + \frac{\partial ...
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80 views

Symmetry and Algebra

I'm trying to get a more concrete idea how symmetry is understood in quantum theories, as broad as possible. Consider a infinitesimal transformation of states in quantum physics of the form: $$ ...
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67 views

Translation Operator on two operators

On my last HW set, we were asked to show that the operator $$\hat T = \exp(-ic\hat p /\hbar)$$ acts as a translation operator ($\hat T^\dagger q\hat T=q+c)$. This was simple to show using commutators ...
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121 views

Commutators with function

I have following exercise: If $[C,D]$ is a c-number and $f(x)$ is a well-behaved function (i.e. all derivatives exist and are finite), show that: $$[C, f(D)]=[C,D]f'(D)$$ where $f'(D) = ...
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122 views

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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92 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) & ...
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45 views

When trying to see what symmetries an operator generates, how do you “decide” what coordinate to apply it to?

Suppose I have $\hat{O}_{1}=-i\hbar\partial_{x}$ then \begin{eqnarray} e^{-i\gamma\hat{O}_{1}/\hbar}x\,e^{i\gamma\hat{O}_{1}/\hbar}=x+\gamma \end{eqnarray} and \begin{eqnarray} ...
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Why is the electric field operator normalized by a volume?

I came across the following definition of the electric field operator: But I am not sure what this $V$, the "volume of a box", is about. It seems to enter the discussion in order to have standing ...
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60 views

Simultaneous eigenket

J. J. Sakurai states in his "Modern Quantum Mechanics", this fact as a theorem ($\pi$ is the parity operator): Suppose $$[H,\pi]=0$$ and $| n>$ is a nondegenerate eigenket of $H$ with ...
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77 views

Non Hermitian Quantum Mechanics

I was just reading about Non-Hermitian Quantum Mechanics dealing with Hamiltonians $H$ that are not Hermitian operators. Then it is unclear that we get orthonormal eigenstates. Now, I was reading a ...
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62 views

Basis spin states

We are given a system of $N$ spin states and the following (non-hermitian) Hamiltonian $$H = \frac{N \hbar \nu}{2M} \sin(\alpha)+ \sum_{i=1}^N \frac{\hbar \omega_i }{2} \sigma_{z,i} + \frac{\hbar \nu ...
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Hamiltonian of a linear chain of atoms and canonical quantization

When we want to re-formalize the Hamiltonian of a linear chain of atoms which has the following form: , we define the ladder operators as: and we use the following relations: to show that we ...
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What is the physical significance of the two integration constants that appear in the ladder operator decomposition of the Quantum Hamiltonian?

If I have a simple one dimensional Hamiltonian of the form \begin{align} H = V - \partial_x^2 \end{align} and if I know one zero energy state solution $H\psi_0=0$ then I can use the Wronskian to ...
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45 views

Finding the expectation value of the annihilation operator with respect to a given state

Using dirac notation we were given a state vector $$|\Psi(t=0)\rangle = A\sum\limits_{q=0}^Q \frac{1}{(q+i)} |\phi_q\rangle$$ Where $\phi$ is part of a complete orthonormal set. I found the ...
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13 views

Split property for type III algebras entails practical separability

I am reading Halvorson's thesis (http://philsci-archive.pitt.edu/346/1/main-new.pdf), however I don't understand a proof at p.50 where he tries to explain why the split property allows a local agent ...
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85 views

Normal ordering

If I understood correctly there are two terms called normal ordering: $:c c^\dagger: = c^\dagger c \hspace{.5cm}$so shifting all creation operators to the left and all annihilation operators to the ...
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53 views

How can I take the Wigner transform of an operator with an absolute value?

I want to be able to find the Wigner transforms of operators of the form $\Theta(\hat{O})$, where $\Theta$ is the Heaviside function and $\hat{O}$ in general depends on both $x$ and $p$. For the ...
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201 views

Transition Between Position and Momentum Basis

I'm having some trouble following pages 55-56 of Sakurai's Modern Quantum Mechanics. We're trying to transfer from position space into momentum space. Here's a quote: Let us now establish the ...
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70 views

Galilean Transform

I tried to solve a problem using two different ways and I had some trouble, the problem is: We define a symmetry transform of the expected value of $\vec{P}$ like this: $$\langle \psi|\vec{P}|\psi ...
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98 views

Momentum and position operators in Schrödinger representation

I was going through some intro notes on path integral (for QFT), and am stuck with this equation for position and momentum in Schrödinger (position) representation, $$ \hat{1} =\int ...
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How does $\bar{r}\times(\bar{\nabla}\times) - \bar{\nabla}\times(\bar{r}\times)$ relate to the orbital angular momentum operator?

When I attempted to calculate the following by hand $$\bar{r}\times(\bar{\nabla}\times\bar{F}) - \bar{\nabla}\times(\bar{r}\times\bar{F}),$$ I noticed some of the terms I extracted looked similar to ...
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112 views

How do I prove the equivalence of chirality and helicity operators acting on a massless Dirac spinor?

I have massless Dirac equation and chirality and helicity operators which are given as $$ \hat {P}_{ch}\Psi = \gamma_{5}\Psi, \quad \hat {P}_{h}\Psi = \frac{(\hat {\mathbf S} \cdot \mathbf ...
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About Divergence in polar coordinates

I've got a conductor in a cylinder shape who is rotating with angular velocity $\omega$ around its axis, that correspond to the $z$ azis I want to calculate the electric field and the density of ...
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64 views

How does linearity of a measurement imply that the commutator of all measured observables are $c$-numbers?

I really don't understand with the linearity conditions I have where this comes from.
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148 views

Time ordering and Fermions

Having time ordering operator for fermions, should it reverse sign if it swaps operators with opposite spin variable? In other words should $T[c_{t_1,\uparrow}c_{t_2,\downarrow}^\dagger]$ return ...
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148 views

QFT basics for Klein-Gordon fields

I am teaching myself QFT from Peskin for next years maths course and I have two questions: What is a c-number? Is it a complex number, and if so why does it mean, ...
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106 views

Explicit evaluation of a radially ordered product

I am trying to understand the application of the operator product expansion to calculate the radially ordered product in the complex plain of $T_{zz}(z)\partial_w X^{\rho}(w)$ which should result in ...
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21 views

Adiabatic Theorem in Terms of Eigenvector Derivatives

The necessary conditions for quantum Adiabatic Theorem validity is usually stated in terms of eigenvalue gaps for parameterized Hermitian matrices, or Hamiltonians. If $H(t)$ is a parameterized ...
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23 views

Fluctuation operator and partial wave

Can someone please explain to me why the expression $[-\Box + U''(\Phi(r))]$ is called the fluctuation operator? I was also wondering how to derive the following for the $l^{th}$ partial wave of the ...
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26 views

Physical significance of Cayley Transform

In the book on Quantum Mechanics by Capri (in Chapter 6), its said that an operator $A$ is self adjoint if the operator, $U$ given by $$ U = (A - i I)(A + i I)^{-1} = -(I+iA)(I-iA)^{-1} = -\text ...
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Identity operator in terms of the energy eigenstates in case of continuous spectrum

Let us confine ourselves to the 1d case. If we define the momentum eigenvector $|k\rangle $ as $$ \langle x |k\rangle = \frac{1}{\sqrt{2\pi}} e^{i k x} ,$$ we have the identity operator decomposed ...
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Precisely when is a matrix representation of Hermitian operator also Hermitian?

I asked a question on math exchange Are properties of linear operators reflected in matrix representations with different output and input basis?. In that question I asked: if I had a Hermitian ...
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59 views

How to calculate the expectation value of position/momentum using path integrals?

We have the formula: \begin{equation} \langle F \rangle = \frac{\int Dx \times F[\phi] exp\{i/\hbar S[\phi]\}}{\int Dx \times exp\{i/\hbar S[\phi]\}} \end{equation} Now, I am wondering how a change ...
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57 views

Does this Hamiltonian have point spectrum?

Consider such a Hamiltonian $$ H = - \frac{1}{2} \frac{\partial^2}{\partial x^2} - F x + V(x) ,$$ with $F$ being some constant, and $V(x)= V(x+L)$ being some periodic potential. Does this ...
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41 views

What is the missing step in this result regarding the creation operators in Fock space?

In the above extract from Simons and Altman: Condensed Matter Field Theory, I am having trouble getting from (2.3) to (2.4) in the case of Fermions (ζ=-1 and the n(subscript i) values are modulo 2). ...
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29 views

Showing a measurement operator has a particular form

I came across an exercise (Ex 1.16) in 'Quantum Measurement and Control' by Wiseman and Milburn that I am having some trouble with. Suppose we have some system $S$ coupled with two meters in states ...