# 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.

204 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
173 views

### Derivation of the Hypersurface Deformation Algebra

Let $({M},{g})$ be a smooth $4d$ spacetime manifold with lorentzian metric $g$ and local coordinates $\xi^{\alpha}$ and let further $({N},{q})$ be a smooth $3d$ manifold with metric $q$ and local ...
283 views

69 views

### Generalized momentum in terms of wavefunction: Is it always $-i\hbar \partial/\partial q$?

I saw this kind of derivation several times in different notes/review/educational articles. (For example https://arxiv.org/abs/1904.06560 or http://wcchew.ece.illinois.edu/chew/course/QMALL20121005....
49 views

### Calculation of commutation relations in the SYK model

I'm reading this paper (https://arxiv.org/abs/1604.07818). And I'm having trouble showing an equality. We consider the following $SL(2,R)$ generators. \begin{align} D=-t\partial_t-\frac{1}{4},\ P=\...
118 views

### Representations of the Poincaré group

I am currently trying to understand the representations of the conformal group. I am following the script by J. D. Qualls. At page 29, the author finds the effect of $L_{\mu\nu}$ by "studying the ...
75 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 ...
106 views

### Raising and Lowering Operators of a Hamiltonian

Lets say that I have a Hermitian Hamiltonian $H$ with a non-Hermitian raising operator operator $A$ which satisfies \begin{equation} [H,A] = \Omega A, \quad \Omega \in \ \mathbb{R}_{>0}. \end{...
126 views

The generalized uncertainty principle can be derived and shown to be this which is fine and rigorous. $\langle ( \Delta A )^{2} \rangle \langle ( \Delta B )^{2} \rangle \geq \dfrac{1}{4} \vert \langle ... 3 votes 0 answers 65 views ### Topological Descent Equation Assume that we have a cohomological field theory, with an odd symmetry generated by an odd operator$Q$and an exact energy momentum tensor$T_{\mu\nu}=[Q,G_{\mu\nu}]$. Then by integrating over an ... 3 votes 0 answers 109 views ### What can be said about the commutator of an operator with itself at different times? In general for some smooth and bounded$\hat{V}$$$\left[\hat{V}(t_1), \hat{V}(t_2) \right] \neq 0 \text{ if } t_1 \neq t_2$$ But what more can be said about commutators of this type? I am ... 3 votes 0 answers 105 views ### Can current and voltage be linked by an uncertainty relation when electrons tunnel through a barrier? Quantum tunneling has been shown to be linked to uncertainty relations for some observables involved in the system. For instance, if we consider electrons tunneling through a potential barrier it can ... 3 votes 1 answer 76 views ### Commutator of net position and net momentum It is well known that $$[\hat{x},\hat{p_x}] = [\hat{y}, \hat{p_y}] = [\hat{z}, \hat{p_z}] = i\hbar$$ But what if instead we wanted to know the commutator of the net displacement$\hat{r} = \sqrt{\...
437 views

In my String theory lecture radial ordering was introduced and I don't understand what it is. My first guess was $$R(A(z)B(w)) = A(z)B(w)\Theta(|z|-|w|) + B(w)A(z)\Theta(|w|-|z|).$$ But then we have ...
235 views

### Dyson series for Hamiltonian with $c$-number commutator

I am trying to derive the evolution operator for a time dependent Hamiltonian which satisfies the commutator $$[H(t_1), H(t_2)]=I f(t_1,t_2)$$ Where $I$ is the identity operator, and $f(t_1,t_2)$ is ...
653 views

130 views

### Use the commutation relation to show that the conjugate momentum acts on eigenstates of $\hat{\Phi}$ as $- i \delta / \delta\phi_a(\mathbf{x})$

This is part (b) of Schwartz's Problem 14.3 in his Quantum Field Theory and the Standard Model textbook. Suppose that we have a real scalar field operator $\hat{\Phi}(x^0,\mathbf{x})$ with conjugate ...
45 views

129 views

### Quantizing first class constraints

Let $\gamma$ denote a first class constraint. Then if there exists a function on phase space $f(q,p)$ for which the Poisson bracket with the constraint does not vanish $\lbrace f, \gamma\rbrace \neq 0$...
169 views

1k views

1k views

### Non-commutativity of operators, simultaneous eigenstates and degeneracies over finite dimensional Hilbert space

Suppose, I have two operators $\hat{A}$ and $\hat{B}$ defined over a finite dimensional Hilbert space $H$. Also assume that both the operators have their own eigenstates that span $H$(existence of ... 