Questions tagged [commutator]
A mathematical construct quantifying the difference in effect of applying two operators in two alternate successions. It is the defining product of a Lie algebra, the efficient underlying description of Lie groups, of use in several areas of physics, most notably quantum field theory.
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$R$-symmetry, commutators and charges in supersymmetry
In my lectures in supersymmetry, i have written the following statement about $R$-symmetry
It is special since it does not commute with supersymmetry. Instead, it obeys $ \left[ R, Q_a \right] = -Q_a ...
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Spin operator acting on meson field operator
In the book "Heavy Quark physics" of Manohar/Wise in chapter 2.5 the meson field operator in the ground state configuration with one heavy and one light quark has been derived. The field ...
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Decomposing operator exponentials [closed]
If I have a commutation relation
$$[A,B] = -2C$$
for complex $k$ and operators $A,B,C$ then how would you deconstruct the operator exponential:
$$e^{kA-k^*B}$$
as the rule
$$e^Ae^B = e^{A+B}$$
only ...
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Observable effects of modified Heisenberg uncertainty for fields
Suppose that a QFT has the following "same-time" commutation relations for two fields:
\begin{equation}
\left[\hat{\psi}(t,\vec{x}),\hat{\Pi}(t,\vec{y})\right] = i\hbar\frac{1}{\pi\sqrt{\...
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Proving quantum operator relationship used in derivation of Radial Schrodinger Equation
In the derivation of the Radial Schrodinger Equation for central potentials, I have seen the following relationship used:
$$
r^2p^2 = L^2+(\bf{r}\cdot\bf{p})^2-i\hbar(\bf{r}\cdot\bf{p})
$$
and I have ...
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Confusion About Time Ordering in the Context Of The LSZ Formula
I'm reviewing some of the derivations of QFT and today I realized I don't fully understand the proof of the LSZ formula given in Srednicki. My issue is with the use of time-ordering specifically. I ...
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Sketching Angular Momentum Vector Field
The following is a question from the book Gravitation by Misner, Thorne, and Wheeler. They define the angular momentum vector in Euclidean Space by the formula $L_j=\epsilon_{jkl}{x^k}\frac{{\partial}}...
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What is the mathematically precise definition of raising and lowering operators?
As a finite dimensional example, we have spin raising and lowering operators with the defining property
$$[S_z, S_+] = \omega S_+,$$
$$\quad \quad \quad \iff [S_z, S_-] = -\omega S_-$$
for some ...
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Is string theory a particular non-commutative field theory (whether the commutator of the position coordinates in string theory is non-zero)?
I am just beginning to study string theory, and am reading a bit of literature. Following this, I have a question which is probably not very well framed:
I want to know whether string theory is a ...
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Commutation of Solutions of the Dirac Field
In page 53 of P&S, they derive equation 3.89, which is the commutator for the dirac field operators (the commutator is derived for pedagogic purposes),
$$[\psi(x), \psi^\dagger(y)] = \int \frac{d^...
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Understanding derivation of commutator for Dirac Field [duplicate]
Im going through Peskin and Shroeder's book on QFT, in the chapter about the Dirac Equation. They are trying to teach why quantizing with the commutator is wrong, and I'm trying to follow the ...
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Multiplication of operators defined by commutation relations
I'm confused about how usual multiplication of operators is uniquely defined. Usually, we only now for physical reasons the anti-/commutation properties of operators so that $AB-BA$ is fixed.
Take for ...
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What is the commutator of the lowering operator $J_-$ and the exponential of $J_z$, arranged so that the lowering operator is always to the right?
I'm trying to perform a computation closely related to the problem below. It's a tricky little problem which I imagine has been tackled in the literature before, but I've had no luck finding it. ...
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Quantum non-demolition condition in Breuer
I am reading "The Theory of Open Quantum Systems" by Francesco Petruccione and Heinz-Peter Breuer and came across this condition for non-demolition measurements (eq 2.237),
$$ \sum_m \text{...
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What is the correct commutation relationship of vector $\boldsymbol{r}$ and vector $\boldsymbol{p}$? [closed]
We choose $\boldsymbol{r}$ and $\boldsymbol{p}$ are 3 dim vector, and try to get $ [\boldsymbol{r},\boldsymbol{p}]$. In spherical coordinate system,
\begin{align*}
[\boldsymbol{r},\boldsymbol{p}] f &...
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Intuitive explanation of commutation relations of Special Conformal Transformations (SCTs) and momentum generators?
Some entries in the conformal algebra can be intuitively justified: the Lorentz algebra says stuff like rotation+rotation=rotation, boost+boost=rotation, etc. The momentum and angular momentum ...
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Exploiting Creation Operator Commutation Relation in HOM Interference Calculation
In this paper where the authors derive the formula for coincidence probability in a Hong-Ou Mandel (HOM) interference effect as a function of time delay $\tau$, they arrive at an equation (15) with ...
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What is the quantum equation of motion for a massless scalar field in $x^\pm = t\pm x$ coordinates?
Consider a massless scalar field in 1+1D. The Heisenberg equation of motion is
$$
\frac{\partial \phi}{\partial t} = i [H, \phi].
$$
Further, since the spacetime translations are implemented by $U =...
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Understanding the field strength tensor $F_{\mu\nu}$ as a commutator
As far as I understand, one way to define the field-strength tensor is by using the commutator of covariant derivatives as follows:
$$-igT^aF^a_{\mu\nu} = [D_\mu, D_\nu]$$
where $T^a$ is a basis for ...
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Density operator commutator in master equation
In Breuer and Petruccione's Theory of Open Quantum Systems Eq. 3.359, they arrive at the master equation for the evolution of a system's density matrix, after interaction with a pointer via a typical ...
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Commutation relation between charge number and phase in superconductors
Consider an isolated Cooper pair box.
The charge number operator is
$$\hat{n} = \sum_{n=-\infty}^{+\infty} n |n\rangle \langle n|$$
where $|n \rangle$ is the state in which $n$ Cooper pairs have ...
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Field strength tensor written as commutator of covariant derivatives in QED
I am currently trying to understand the derivation of the relation
$$
\begin{equation}
F_{\mu\nu} = \frac{1}{iq}[D_{\mu}, D_{\nu}]\tag{1}\label{eq1}
\end{equation}
$$
in QED and I have trouble with ...
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Causality for gauge dependent operators in quantum field theories
Suppose that $\mathcal{A}_{ij...}(x)$ and $\mathcal{B}_{ij...}( x')$ are two gauge dependent (so non-observable) operator in some theory. If they are spacelike, should I impose the causality ...
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Conjugate observables - can the commutation relations be generalised?
Conjugate variables are variables that are Fourier transforms of one another (that is they are Fourier transform duals) and consequently have an uncertainty relation existing between them. In quantum ...
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Do different bases of Fock space commute?
$\newcommand\dag\dagger$
Suppose we have a Fock space $\mathcal{F}$ with two different bases of creation and annihilation operators $\{a_\lambda, a^\dag_\lambda\}$ and $\{a_{\tilde \lambda}, a^\dag_{\...
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The commutation relations of photon and gluon?
In QED, the photon field has the following commutation relations:
\begin{equation}
[A^{\mu}(t,\vec{x}),A^{\nu}(t,\vec{y})]=0, \tag{1}
\end{equation}
where $A^{\mu}(t,\vec{x})$ is the photon filed. ...
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Does the creation operators for photons with different polarization commute?
Let $\hat{a}^{\dagger}_{\sigma}$ be the creation operator of a photon with the polarization $\sigma $ towards some reference. What are the commutator relations for the creation operators of a photon ...
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Definition of the Conformal algebra generators
In the textbook "Introduction to Conformal Field Theory" by Blumenhagen and Plauschinn (2009), Section 2.1, the generators of the Lie algebra corresponding to the conformal group for the ...
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Is it possible to derive Schrödinger's equation from Hamilton's equations?
Accepting the postulates of quantum mechanics, so promoting the classical dynamical variables to operators with appropriate commutation relations, is it possible to "derive" Schrödinger's ...
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Canonical commutation relation in QFT
The canonical commutation relation in QFT with say one (non-free) scalar real field $\phi$ is
$$[\phi(\vec x,t),\dot \phi(\vec y,t)]=i\hbar\delta^{(3)}(\vec x-\vec y).$$
Is this equation satisfied by ...
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Possible ambiguities of quantization
Quantization means to replace $p$ (the momentum) in the expressions of classical physical quantities with $-i\hbar\nabla$, so we get an operator belonging to each physical quantity. However, an ...
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Graded cyclic properties in tensor calculus formalism of supergravity
I am trying to understand the chapter 4 of https://arxiv.org/abs/hep-th/0204035. I want to obtain equation 4.19 in this article.
First let me summarized some equations we need
Denoting the gauge ...
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Commutation in the Local Gauge Transformations
Let's say that I have a Unitary Local Gauge Transformation $U$, in which the Lie Generators are $T$:
$$ \partial_{\mu} U = \partial_{\mu} e^{-i T^{a} \alpha_{a}(x)} = U \partial_{\mu} \left( -i T^{a} \...
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Deriving the properties of the Dirac matrices
I am working on the properties of the Dirac matrices, but I cannot figure out the derivations.
For example, on proving $\gamma^\mu {\not}{a} \gamma_\mu = -2{\not}{a}$, we first prove that $\gamma^\mu \...
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Taylor condition on the general formula for momentum commutator [closed]
My quantum homework asked me the following question:
Prove that for any $f(x)$ such that $f$ admits a Taylor expansion, the following is true:
$$[f(x), \hat{p}] = i\hbar\frac{\mathrm{d}f}{\mathrm{d}x}...
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Time ordering for a time-dependent Hamiltonian in Path integral derivation
I am currently taking a class on Quantum Field Theory. The propagator was defined as:
$$K(x,t;x',t') = \langle x|\hat{T}e^{{\frac{-i}{\hbar}\int_{t}^{t'}dtH(t)}}|x\rangle$$
where, $\hat{T}$ is the ...
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What does "not applying the CCR" mean exactly?
I've seen mentioned in a number of posts that some relations do or do not apply depending on whether one is "applying the CCR". For example,
In Relationship between normal-ordered vacuum ...
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An interpretation for Propagator $D(x-y)$
When I learn QFT I always see that when we consider the causality problem in QFT, at first we may try to compute the propagator $D(x-y)$ for spacelike distance $(x-y)^2<0$, which is nonzero. An ...
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Do gamma matrices commute with 4-vectors?
One of my exercises was to prove the identity
$$\gamma^\mu\displaystyle{\not}a\gamma_\mu=-2\displaystyle{\not}a.$$
Which is trivial if $\gamma^\mu a_\nu=a_\nu \gamma^\mu$, as follows
$$\gamma^\mu\...
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Knowing all spin components at the same time [duplicate]
You can't know all spin components simultaneously due to the commutation relation (& Heisenberg's uncertainty principle):
$[S_x, S_y] = i\hbar S_z$
But what if you know that $S_z=0$? Then that ...
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Correlation functions of exponentials of fields
I've been trying to solve for scattering amplitudes for 4 graviton scattering in string theory. However, while going through Schwarz, Witten and Green book for string theory, I come across the ...
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Marginal of Exponential Operator in terms of $x$ and $p$
I am effectively looking to compute the marginal,
$\langle x| \exp(\hat x\hat p ) | p \rangle$. Since $[\hat x, \hat p] = i$, it is not simply solved by trivially acting $\hat x$ on to the state on ...
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General bosonic commutator result
I'm considering these functions $f,g,h,j$, which all share the same form, of the boson creation and annihilation operators:
$$ f(\hat{a}^{\dagger} ,\hat{a}) = \sum_{n,m} c_{n,m}(\hat{a}^{\dagger})^n (\...
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Proof that commuting Dirac fields violate causality
What is the proof that commuting Dirac fields violate causality? All sources I could find just quote this result, but I couldn't find an explicit derivation anywhere. In particular, the case I am ...
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Does the anticommutator of two spinors affect the transpose of their product?
My lecture notes claim that for an anticommutation relation
$$[ \psi_{\mu}(\bf{x},t),{\psi_{{\nu}}^{*}}(\bf{y},t)] = \delta_{\mu \nu} \delta^3(\bf{x}-\bf{y})$$
between two spinors, the transpose of ...
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How does $[J^2, J_+] = 0$, but $ | l, m \rangle $ is not an eigenstate of $J_+$? [duplicate]
I am currently working on the angular momentum part (chapter 3) of Sakurai's QM 2nd. From eq. 3.159 and eq. 3.154, the book says $[J^2, J_+] = 0$ and $[J^2, J_z] = 0$, but I don't understand why $| l, ...
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What do you get when you Taylor expand a Magnus expansion?
The Magnus expansion and Dyson series are very similar to each other, in that they both give a way to approximate a time-evolution operator as a series expansion
$$U(t) = \mathcal{T}\left(\exp\left[-i\...
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Constant of Motion in Quantum Mechanics for explicit time-dependent Operators
I was studying constants of motion in quantum mechanics, and at first, I don't understand the condition to be a constant of motion. Generally, the temporal variation of an operator $A$ is given by the ...
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How can interacting field operators in $2D$ still satisfy the canonical commutation relation?
Free fields in any dimensions are well-known to be Gaussian, act on the Fock space and satisfy the canonical commutation relations.
By definition, interacting field operators are NOT such cases, as ...
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Does the Hamiltonian always commute with the Time Evolution Operator?
The time evolution operator $U(t, t_0)$ is given as the solution of the equation
$$ i\hbar \dfrac{\text{d}}{\text{d}t} U(t, t_0) = HU(t, t_0)$$
whether or not the system is conservative. When the ...
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