Questions tagged [anticommutator]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
1answer
86 views

Why are Grassmann variables the classical limit of fermions?

In many texts the anti-commutation relations for fermions are given as $$\{ \bar{\psi}^\alpha (\vec{x}), \psi^\beta(\vec{y}) \} = \delta^{\alpha\beta} \delta(\vec{x} - \vec{y})$$ $$\{ \psi^\alpha (\...
0
votes
0answers
25 views

Commutation relations in QFT [duplicate]

So I have just started learning QFT. So you take a classical field and turn the degrees of freedom into operators. All fine, just like normal quantum. However I am confused about the commutation ...
2
votes
0answers
71 views

How are anticommuting fields ($\eta \chi = -\chi \eta$) “forced upon us” by representation theory of $SO(d-1,1)$?

I would like to know if anticommuting fields (which physicists use as fermions) emerge naturally from the spin representation theory of $SO(d-1,1)$. Is the fact that spinor fields anticommute a ...
0
votes
1answer
24 views

Anticommutativity of an anticommutator of supercharges

In this paper, equation 38 gives the ${\cal N}=2$ Super-Poincare (extended with the central extension $\mathcal{Z}$). The anticommutation relation of the two different supercharges is given as: $$\{Q^...
0
votes
1answer
39 views

Anti-Commutator of derivatives of Grassmann variables

How do I evaluate the anti-commutator of $\frac{\partial}{\partial\chi}$ and $\frac{\partial}{\partial\eta}$ when both $\chi$ and $\eta$ are Grassmann variables?
1
vote
3answers
118 views

Why doesn't the anticommutator $\{x,p_x\}$ have an unique value?

The commutator of position and momentum, $[x,p_x]$, has a unique value given by $i\hbar$. Why doesn't the anticommutator $\{x,p_x\}$ also have a definite value?
0
votes
1answer
25 views

Symmetry in Fock-space 2-body interaction

The simplest two body interaction term for fermions is $$H = \sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l$$ and I'm trying to determine the symmetries on $U$. Unfortunately I keep getting ...
-2
votes
3answers
103 views

Heisenberg Uncertainty Principle derivation question [closed]

So I'm rading Shankaar's book and got stuck in this place. $$ (\Delta \Omega)^{2}(\Delta \Lambda)^{2} \geq \frac{1}{4}\left\langle\psi\left|[\hat{\Omega}, \widehat{\Lambda}]_{+}\right| \psi\right\...
0
votes
0answers
66 views

What is the QFT state with two distinguishable fermions present?

I want to describe a system with two non-interacting and definitely different fermions, say an electron neutrino, $\nu_e$, and an electron, $e^-$. The state describing a single electron is given ...
1
vote
1answer
50 views

Explicit quantization of free fermionic field

The canonical quantization of a scalar field $\phi(x)$ can explicitly be realized in the space of functionals in fields $\phi(\vec x)$ (here $\vec x$ is spacial variable) by operators \begin{eqnarray} ...
3
votes
1answer
110 views

Canonical Quantisation vs the Dirac Field, why does it even work?

Using the "Dirac Prescription", we can preserve the format of a differential equation in its QM form. If we define the canonical variables s.t. they have the same commutation relations times $i$ as ...
0
votes
1answer
57 views

Proving identity $\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu}$

In the lecture notes accompanying a course I'm following, it is stated that $$\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu} $$ Yet when I try to prove this,...
1
vote
1answer
43 views

Commutation relations of symmetry generators in SUSY

It is well known that the generators $$ Q_\alpha = \frac{\partial}{\partial \theta^\alpha} - i \sigma^\mu_{\alpha \dot \beta} \bar{\theta}^\dot{\beta} \partial_\mu $$ and $$ \bar{Q}_\dot{\alpha} = -\...
0
votes
1answer
151 views

Anti-commutation relations in annihilation operators

It is claimed that $$\{c_\alpha,c_\beta \} = c_\alpha c_\beta + c_\beta c_\alpha = 0$$ where $c_\alpha$ and $c_\beta$ are the fermionic annihilation operators in second quantization. Why is that ...
1
vote
3answers
402 views

Is there an anticommutator relation for orbital angular momentum?

So I know that there are commutator relations for $L$ such as $[L_x,L_y] = i\hbar L_z$, but is there a relation for the anticommutator? For example, $L_xL_y + L_yL_x$?
1
vote
1answer
99 views

Anticommutation relation different specie/type of fermion

Suppose we have two distinct fermions, say $X$ is Dirac, $Y$ is Majorana, part of different irreps of some Gauge group (e.g. SM). Alternatively, consider a lepton $l=l_L+l_R$ and a Majorana neutrino $...
1
vote
2answers
137 views

How to prove the translation generator commutes with the spinors in SUSY algebra?

I was reading Modern Supersymmetry by John Terning, the book starts with SUSY algebra and says $$ \left[ P_{\mu} , Q_{\alpha} \right] = \left[ P_{\mu} , Q_{\alpha}^{\dagger} \right] = 0 $$ I am ...
2
votes
2answers
143 views

How to change a commutator of SUSY super-charges into an anti-commutator?

I would like to understand an apparently rather simple calculation which checks the closure of the Supersymmetry algebra via the commutator of 2 supersymmetric variations of the type: $$\delta \phi = ...
1
vote
1answer
65 views

Problems with anti-commutator between fermionic ladder operators

I am trying to build the fermionic coherent state formalism in conformance with the grassmann conventions used in the book "Mirror Symmetry", relation (9.20), where the fermionic integration is ...
2
votes
1answer
2k views

Angular and linear momentum operators' commutation

Do linear and angular momentum operators commute? If I use the canonical commutation relations I get that they commute. Say, for $x$-component, $[p_x, L_x] = p_x y p_z - y p_z p_x - p_x z p_y + z p_y ...
0
votes
0answers
54 views

Anticommutation relation for the exponential field of the bosonic field

In 1+1 dimensions, the massless KG equation has the general solution $$\phi(x,t)=\int_{-\infty}^{\infty}dp/(4\pi E_p)[a_pe^{i(px-E_pt)}+a^{\dagger}_pe^{-i(px-E_pt)}]$$ where $E_p^2=p^2$. The operator ...
1
vote
3answers
161 views

What does self-closing bra-ket mean in Robetson-Schrodinger Uncertainty Relation?

I was reading: https://en.wikipedia.org/wiki/Heisenberg%27s_uncertainty_principle#Robertson–Schrödinger_uncertainty_relations Where an inequality is presented: $$ \sigma_A \sigma_B = | \frac{1}{2} \...
3
votes
4answers
2k views

Proof of the Anti-Commutation Relation for Gamma Matrices from Dirac Equation

My textbook on QFT says that the Dirac equation can be used to show the following relation: $$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$$ I have searched around and unable to find how to prove this ...
0
votes
1answer
60 views

Anticommutator expression

I need to show that this expression is contradictory. The is no more information is given for $\hat{b}$. $$\hat{b}^{\dagger}\hat{b}+\hat{b}\hat{b}^{\dagger}=-I$$
2
votes
1answer
335 views

Why must fermion fields anticommute and bosons commute?

Fermion fields must satisfy anticommutation relation. But why? I know that unless they anti-commute the Pauli exclusion principle cannot be satisfied. But is there some other deeper/fundamental ...
1
vote
1answer
212 views

What's the reasoning behind propagators definitions (specifically fermionic propagators)

I'm studying QFT by David Tong's lecture notes. When he discusses causility with real scalar fields, he defines the propagator as $$D(x-y)=\left\langle0\right|\phi(x)\phi(y)\left|0\right\rangle=\int\...
0
votes
1answer
66 views

Do half integer spin fields commute or anti-commute with spin integer fields?

What are the fundamental commutation/anti-commutation relations between half integer and integer spin fields? For instance, in QED do we have \begin{equation} [\psi(x),A^{\mu}(y)]=0 \end{equation} or \...
0
votes
0answers
108 views

Anti-commutator version of Zassenhaus formula

The Zassenhaus formula goes like $$ e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y] + 3[[[X,Y],Y],...
0
votes
1answer
348 views

Quantizing the Dirac field using commutation relations leads to an unbounded Hamiltonian?

If we were to quantize the Dirac field using commutation relations instead of anticommutation relations we would end up with the Hamiltonian $$ H = \int\frac{d^3p}{(2\pi)^3}E_p \sum_{s=1}^2 ...
4
votes
2answers
415 views

Relation between spinors and anticommutation relation of fermions

I read that the state of a pair of particles is the tensor product of the single states of both, and you will get a wavefunction with the parameters of both, if you swap the parameters you will get a ...
5
votes
2answers
497 views

Why must the Bogoliubov transform preserve anticommutation relations?

$\mathbf{Background}$: In my research I am studying the Ising model, expressed in terms of Jordan-Wigner fermions: $$ H = \sum_{j=1}^n(c_j - c_j^\dagger)(c_{j+1} + c_{j+1}^\dagger) + \lambda c_jc_j^\...
-1
votes
1answer
149 views

Anti-commutative Hermitian operators in an infinite dimensional Hilbert space

An example of a pair of anti-commutative Hermitian operators in a finite Hilbert space is $\sigma_x$ with $\sigma_z.$ Indeed $\sigma_z\sigma_x=i\sigma_y$, whereas $\sigma_x\sigma_z=-i\sigma_y$. My ...
1
vote
0answers
286 views

Numerically Calculate expectation of $xp+px$?

I'm curious if there is a quick way to numerically calculate $\langle xp + px \rangle$ if we had the density function of our system. For example, if for x we can take the density function in the x-...
2
votes
2answers
103 views

Can the necessity of using anti-commutators for Dirac fields and commutator for Klein-Gorden be deduced from the field equations?

We all learned to use the commutator for quantizing the KG field and the anti-commutator for the Dirac field. We are told (which is correct) so that KG-excitations are bosons and Dirac-excitations ...
1
vote
2answers
10k views

Properties of anticommutators [closed]

Do anticommutators of operators has simple relations like commutators. For example: $$[AB,C]=A[B,C]-[C,A]B.$$ But I don't find any properties on anticommutators. Do same kind of relations exists ...
1
vote
2answers
195 views

Canonical quantization of bosons

During my studies on QFT a fundamental question occurred concerning the canonical quantization. In our course, we mentioned that: "The canonical quantization of a field with values in the complex ...
2
votes
1answer
2k views

Confusion about slash notation

I am confused about the slash notation and especially taking the square of a slashed operator. Defining $\displaystyle{\not} a \, = \, \gamma^\mu a_\mu$ we have $\,\,$ $\displaystyle{\not} a \...
0
votes
1answer
222 views

Fermion anti-commutation relations

The fermion anti-commutation relations are given as $$\{\psi_{\alpha}({\bf x},t),\psi_{\beta}^{\dagger}{(\bf x'},t)\} = \delta_{\alpha,\beta} \, \delta({\bf x} - {\bf x'}).$$ I am interested in ...
0
votes
1answer
77 views

Field operator anti-commutator relation

For the field operators (fermions) $$\hat{\Psi}^\dagger_\sigma(x) = \dfrac{1}{\sqrt{V}}\sum_k e^{-ikx}~\hat{a}^\dagger_{k,\sigma}$$ $$\hat{\Psi}_\sigma(x) = \dfrac{1}{\sqrt{V}}\sum_k e^{ikx}~\hat{a}...
2
votes
2answers
313 views

Second Quantization: Do fermion operators on different sites HAVE to anticommute?

In second quantization, we assume we have fermion operators $a_i$ which satisfy $\{a_i,a_j\}=0$, $\{a_i,a_j^\dagger\}=\delta_{ij}$, $\{a_i^\dagger,a_j^\dagger\}=0$. Another way to say this is that $$ ...
7
votes
2answers
827 views

Why do we use the anticommutation relation for particle-hole and chiral symmetries?

In physics we say that a quantity is conserved, if its operator commutes with Hamiltonian. For example, in condensed matter systems, when the momentum $k$ commutes with the Hamiltonian $H$ as $[H,k]=...
6
votes
1answer
464 views

Can I always find an unitary operator $B$ such that $\{A,B\}= 0$ for a given, unitary operator $A$?

Considering an arbitrary unitary operator $A$, what is the least criteria this operator must satisfy in order that it is possible to find at least another unitary operator $B$ that anti-commutes with ...
8
votes
4answers
1k 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 ...
6
votes
3answers
2k views

The anticommutator of $SU(N)$ generators

For the Hermitian and traceless generators $T^A$ of the fundamental representation of the $SU(N)$ algebra the anticommutator can be written as $$ \{T^A,T^{B}\} = \frac{1}{d}\delta^{AB}\cdot1\!\!1_{d} +...
2
votes
1answer
429 views

Derivation of fermion anticommutation rule

How one might derive the fermionic anticommutation relations? For bosonic particles, there is no ordering issue, and its commutation relation could be easily derived. However, for fermion, is there ...
3
votes
1answer
393 views

Minus sign in the time ordering operator

The time ordering operator is usually defined as $$\mathcal{T} \left\{A(\tau) B(\tau')\right\} := \begin{cases} A(\tau) B(\tau') & \text{if } \tau > \tau', \\ \pm B(\tau')A(\tau) & \text{if ...
3
votes
1answer
436 views

Wrong sign anticommutation relation for the Dirac field?

Consider the Dirac Lagrangian $$\mathcal{L}=\psi ^{\dagger }\gamma ^{0}\left( \mathrm{i}\gamma ^{\rho }\partial _{\rho }-m\right) \psi .$$ The conjugate momenta to $\psi ^{a}$ are defined, as usual, ...
0
votes
2answers
126 views

Explanation on anticommutation relations

Setup Given two states: $|K\rangle=a_i^+a_j^+|\rangle$ and $|L\rangle=a_k^+a_l^+|\rangle$. Evaluating the overlap: $\langle K|L\rangle=\langle|a_ja_ia_k^+a_l^+|\rangle$ Introducing: $a_ia_k^+=\delta_{...
2
votes
1answer
88 views

Anticommutator difference [closed]

What is the value of this difference of anticommutators $$\{x^2,p^2\}-(\{x,p\}^2)/2$$ if the commutator $$[x,p]=i\hbar ~?$$ I have tried and obtained a value $$-3\hbar^2/2 - 2i\hbar px.$$ But the ...
0
votes
2answers
348 views

A problem with indistinguishable fermions and the order of applying operators

This question comes in consequence of another one. I want to stress a problem that none of the answers addressed it. For making my problem more understandable let me first remind a well-known state, ...