# 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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### 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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### 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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### The commutation relations of photon and gluon?

In QED, the photon field has the following commutation relations: $$[A^{\mu}(t,\vec{x}),A^{\nu}(t,\vec{y})]=0, \tag{1}$$ 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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### 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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### 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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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 ...