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.

Filter by
Sorted by
Tagged with
97
votes
6answers
24k views

What is the physical meaning of commutators in quantum mechanics?

This is a question I've been asked several times by students and I tend to have a hard time phrasing it in terms they can understand. This is a natural question to ask and it is not usually well ...
95
votes
6answers
14k views

Trace of a commutator is zero - but what about the commutator of $x$ and $p$?

Operators can be cyclically interchanged inside a trace: $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero: $${\rm Tr} ([A,B])~=~0.$$ But what about ...
57
votes
5answers
28k views

What is the connection between Poisson brackets and commutators?

The Poisson bracket is defined as: $$\{f,g\} ~:=~ \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\...
49
votes
7answers
79k views

What is the Physical Meaning of Commutation of Two Operators?

I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable (...
44
votes
2answers
6k views

In QFT, why does a vanishing commutator ensure causality?

In relativistic quantum field theories (QFT), $$[\phi(x),\phi^\dagger(y)] = 0 \;\;\mathrm{if}\;\; (x-y)^2<0\,.$$ On the other hand, even for space-like separation $$\phi(x)\phi^\dagger(y)\ne0\,.$...
42
votes
6answers
3k views

What is the “secret ” behind canonical quantization?

The way (and perhaps most students around the world) I was taught QM is very weird. There is no intuitive explanations or understanding. Instead we were given a recipe on how to quantize a classical ...
37
votes
1answer
6k views

Understanding Poisson brackets

In quantum mechanics, when two observables commute, it implies that the two can be measured simultaneously without perturbing each other's measurement results. Or in other words, the uncertainty in ...
36
votes
4answers
11k views

How does non-commutativity lead to uncertainty?

I read that the non-commutativity of the quantum operators leads to the uncertainty principle. What I don't understand is how both things hang together. Is it that when you measure one thing first and ...
33
votes
7answers
5k views

Is there something behind non-commuting observables?

Consider a quantum system described by the Hilbert space $\mathcal{H}$ and consider $A,B\in \mathcal{L}(\mathcal{H},\mathcal{H})$ to be observables. If those observables do not commute there's no ...
24
votes
2answers
2k views

Motivating Complexification of Lie Algebras?

What is the motivation for complexifying a Lie algebra? In quantum mechanical angular momentum the commutation relations $$[J_x,J_y]=iJ_z, \quad [J_y,J_z] = iJ_x,\quad [J_z,J_x] = iJ_y$$ become, on ...
24
votes
2answers
4k views

What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?

I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following: $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$. $|p\rangle$ is an eigenvector of ...
18
votes
5answers
4k views

Quantizing a complex Klein-Gordon Field: Why are there two types of excitations?

In most references I've seen (see, for example, Peskin and Schroeder problem 2.2, or section 2.5 here), one constructs the field operator $\hat{\phi}$ for the complex Klein-Gordon field as follows: ...
16
votes
3answers
1k views

Dequantizing Dirac's quantization rule

In this blog post, Lubos Motl claims that any commutator may be shown to reduce to the classical Poisson brackets: $$ \lim \limits_{\hbar \to 0} \frac{1}{i\hbar} \left[ \hat{F}, \hat{G} \right] =...
16
votes
2answers
669 views

Why is the uncertainty principle not $\sigma_A^2 \sigma_B^2\geq(\langle A B\rangle +\langle B A\rangle -2 \langle A\rangle\langle B\rangle)^2/4$?

In Griffiths' QM, he uses two inequalities (here numbered as $(1)$ and $(2)$) to prove the following general uncertainty principle: $$\sigma_A^2 \sigma_B^2\geq\left(\frac{1}{2i}\langle [\hat A ,\hat B]...
15
votes
3answers
4k views

Is Planck's Constant Really a Constant?

I am going through Groenewold's theorem and in his book: On The Principles of Elementary Quantum Mechanics, page 8, eq. 1.30: $$[\mathbf{p}, \mathbf{q}]=1\left(\text { i.e. } \mathbf{p q}-\mathbf{...
15
votes
1answer
3k views

Validity of Bogoliubov transformation

In condensed matter physics, one often encounter a Hamiltonian of the form $$\mathcal{H}=\sum_{\bf{k}} \begin{pmatrix}a_{\bf{k}}^\dagger & a_{-\bf{k}}\end{pmatrix} \begin{pmatrix}A_{\bf{k}} &...
15
votes
4answers
3k views

Fermions, different species and (anti-)commutation rules

My question is straightforward: Do fermionic operators associated to different species commute or anticommute? Even if these operators have different quantum numbers? How can one prove this fact in a ...
15
votes
2answers
11k views

Mutual or same set of eigenfunctions if two Hermitian operators commute

If two operators commute, do they have "a mutual set of eigenfunctions", or "the same set of eigenfunctions"? My quantum chemistry book uses these as if they are interchangeable, but they do not seem ...
14
votes
2answers
16k views

Commutator with exponential $[A, \exp(B)]$

How can I tell if $A$ and $\exp(B)$ commute? For $[A, B]$ it's simply $AB-BA$ and for $[\exp(A), \exp(B)]$ I think it'd be $\exp(A)\exp(B) - \exp(B)\exp(A) = \exp(A+B) - \exp(B+A) = 0$. Update: it's ...
14
votes
2answers
2k views

A question about causality and Quantum Field Theory from improper Lorentz transformation

Related post Causality and Quantum Field Theory In Peskin and Schroeder's QFT p28, the authors tried to show causality is preserved in scalar field theory. Consider commutator $$ [ \phi(x), \phi(y) ]...
14
votes
1answer
464 views

Does a complete set of commuting observables always exist?

The Wikipedia article for complete set of commuting observables (CSCO) outlines a method for finding a CSCO. However, it does not suggest why such a set should exist. On one hand, I think it is very ...
14
votes
2answers
4k views

What exactly is the connection between the Jacobi and Bianchi identities?

While reviewing some basic field theory, I once again encountered the Bianchi identity (in the context of electromagnetism). It can be written as $$\partial_{[\lambda}\partial_{[\mu}A_{\nu]]}=0.$$ ...
13
votes
4answers
19k views

How to construct the radial component of the momentum operator?

I'm having trouble doing it. I know so far that if we have two Hermitian operators $A$ and $B$ that do not commute, and suppose we wish to find the quantum mechanical Hermitian operator for the ...
13
votes
2answers
336 views

Why does $\hbar$ appear twice in the axioms of QM?

Physical theories have dimensionful constants. Each constant can be found via measurement, by fitting some equation to data. Mathematically, you would expect each constant to be "defined" in this way ...
13
votes
1answer
10k views

Momentum as Generator of Translations

I understand from some studies in mathematics, that the generator of translations is given by the operator $\frac{d}{dx}$. Similarly, I know from quantum mechanics that the momentum operator is $-i\...
13
votes
2answers
631 views

What is the mistake in calculating such a commutator? [duplicate]

$B$ is an Hermitian operator in Hilbert space, and $|b\rangle$ is the eigenstate of $B$. We can have $[A, B] = 1$ where A is arbitary operator. Then we can calculate as below: \begin{align} &\...
12
votes
3answers
4k views

Does the canonical commutation relation fix the form of the momentum operator?

For one dimensional quantum mechanics $$[\hat{x},\hat{p}]=i\hbar. $$ Does this fix univocally the form of the $\hat{p}$ operator? My bet is no because $\hat{p}$ actually depends if we are on ...
12
votes
2answers
2k views

Commutator not transitive

I noticed the following: $$[L_{+},L^2]=0,\qquad [L_{+},L_3]\neq 0,\qquad [L^2,L_3]=0.$$ This would suggest, that $L^2,L_+$ have a common system of eigenfunctions, and so do $L^2,L_3$, but $L_+,L_3$ ...
12
votes
3answers
2k views

Do fields describing different particles always commute?

Is it true that field operators describing different particles (for example a scalar field operator $\phi (x) $ and a spinor field operator $\psi (x) $) always commute (i.e. $ [\phi (x), \psi (y) ]=0, ...
12
votes
4answers
6k views

Binomial expansion of non-commutative operators

I would like to determine the general expansion of $$(\hat{A}+\hat{B})^n,$$ where $[\hat{A},\hat{B}]\neq 0$, i.e. $\hat{A}$ and $\hat{B}$ are two generally non-commutative operators. How could I ...
12
votes
2answers
393 views

Operators “carrying” momentum and particle number

TL;DR: What does it mean for a operator in QFT to "carry momentum"? During a QFT lecture (discussing the real scalar field) my Prof. stated that the operator $$P^\mu := \int \frac{d^d\vec{p}}{(2\pi)^...
12
votes
2answers
4k views

Why are anticommutators needed in quantization of Dirac fields?

Why is the anticommutator actually needed in the canonical quantization of free Dirac field?
11
votes
2answers
902 views

Commutator of two functions

I want to compute the commutation relation between a function of momentum and a function of space in Quantum Mechanics. I know the commutation relation between momentum and a function of space but how ...
11
votes
3answers
5k views

Eigenstate of position+momentum?

I'm studying Quantum Mechanics on my own, so I'm bound to have alot of wrong ideas - please be forgiving! Recently, I was thinking about the quantum mechanical assertion (postulate?) that states with ...
11
votes
3answers
7k views

Why Don't the Ladder Operators Commute?

I have two problems with ladder operators. The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very ...
11
votes
4answers
20k views

Commutator of Dirac gamma matrices

Quick question...For some reason I'm having trouble finding an identity or discussion for the commutator of the gamma matrices at the moment...i.e $$\gamma^u\gamma^v-\gamma^v \gamma^u$$ but I am not ...
11
votes
3answers
9k views

What is the commutator of an operator and its derivative?

Is it possible to calculate in a general way the commutator of an operator $O$ which depends on some variable $x$ and the derivative of this $O$ with respect to $x$? $${O}={O}(x)\\ \left[\partial_x{O}(...
11
votes
5answers
403 views

Does an Operator that neither commutes with $\hat{X}$ or $\hat{P}$, nor can be expressed as a “function” of $\hat{X}$ and $\hat{P}$ make sense?

When you come from classical hamiltonian mechanics (which is based on the phase space), observables are introduced as functions $f$ on the phase space $(q, p)$. There can't be a classical observable ...
11
votes
2answers
2k views

How to know if a set of commuting observables is complete?

We define a complete set of commuting observables as a set of observables $\{A_1,\ldots, A_n\}$ such that: $\left[A_i, A_j\right]=0$, for every $1\leq i,~j \leq n$; If $a_1,\ldots, a_n$ are ...
11
votes
2answers
629 views

How to replace $T$-product with retarded commutator in LSZ formula?

I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246: Here, they consider the elastic scattering of particle $A$ off particle $B$: ...
10
votes
2answers
2k views

Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are $[D,P_{...
10
votes
1answer
3k views

Commutator of Lorentz boost generators : visual interpretation

I have always struggled to visualize the correctness of the commutation relation for the generators of the boost in the Lorentz group. We have $$[K_i,K_j] = i \epsilon_{ijk} L_k$$ I fail to picture ...
9
votes
3answers
5k views

Canonical Commutation Relations

Is it logically sound to accept the canonical commutation relation (CCR) $$[x,p]~=~i\hbar$$ as a postulate of quantum mechanics? Or is it more correct to derive it given some form for $p$ in the ...
9
votes
2answers
2k views

Symmetry in quantum mechanics

My professor told us that in quantum mechanics a transformation is a symmetry transformation if $$ UH(\psi) = HU(\psi) $$ Can you give me an easy explanation for this definition?
9
votes
3answers
14k views

Proof of Canonical Commutation Relation (CCR)

I am not sure how $QP-PQ =i\hbar$ where $P$ represent momentum and $Q$ represent position. $Q$ and $P$ are matrices. The question would be, how can $Q$ and $P$ be formulated as a matrix? Also, what is ...
9
votes
2answers
15k views

Fundamental Commutation Relations in Quantum Mechanics

I am trying to compile a list of fundamental commutation relations involving position, linear momentum, total angular momentum, orbital angular momentum, and spin angular momentum. Here is what I have ...
9
votes
4answers
1k views

Source of Ladder Operators

This might be a silly question but I am curious as to where the ladder operators in quantum mechanics come from. For example, in introductory texts on quantum mechanics, they try to solve the ...
9
votes
2answers
529 views

Quantum mechanics with multiple values of $\hbar$

The quantity $\hbar$ appears in quantum mechanics by the canonical commutation relation $$[x, p] = i \hbar.$$ Would it be sensible to quantize different conjugate variable pairs with different values ...
9
votes
2answers
434 views

Renormalization and canonical commutation relations

My question is whether canonical commutation relations hold for renormalized quantum fields. Below I show reasoning which caused by doubts. Consider a relativistic scalar QFT. We have spectral ...
8
votes
5answers
8k views

Commutator algebra in exponents

Considering $X$ and $Y$ such that $[X,Y]=\lambda$, which is complex, and $\mu$ is another complex number, prove: $$e^{\mu(X+Y)}=e^{\mu X} e^{\mu Y} e^{-\mu^2\lambda/2}$$ My attempt (so far) is: ...

1
2 3 4 5
20