Questions tagged [anticommutator]
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183 questions
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Microcausality in Fermions
We can show that for the Dirac field, the anti-commutator between the field and its adjoint vanishes for space-like separated points. However, for causality we need to show that the commutator instead ...
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Minus sign of anticommuting Dirac fields
I have a conceptual question regarding the implicit assumption, which appears in QFT books. When we deal with Dirac bilinears, e.g. $\bar\psi(x)\psi(x)$, and perform some manipulations requiring ...
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How can I derive the group of canonical/bogoliubov transformations for bosons and fermions?
It is often stated (and never derived) that the group of bosonic canonical transformations is $\text{Sp}(2N, \mathbb{R})$ and the group of fermionic canonical transformations is $O(2N, \mathbb{R})$. ...
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Canonical quantisation of conjugate Dirac field
This may be a stupid question to ask. For Dirac field, we know the Lagrangian $${\cal L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi \tag{1}$$ is not symmetric in $\psi$ and its conjugate field $$\...
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Supersymmetric Hamiltonian for particle on a line with internal degrees of freedom
I was going through David Tong's Supersymmetric Quantum Mechanics
https://www.damtp.cam.ac.uk/user/tong/susyqm.html
On page 8, (1.7), is the expression for the Hamiltonian correct? When I calculated, ...
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How to get the anticommutator $\{b_m,c_n\}$?
Suppose $b(z)$ and $c(z)$ are holomorphic fermionic ghost fields with conformal dimension 2 and -1, respectively, with mode expansions $$b(z)=\sum_n \frac{b_n}{z^{n+2}}$$ and $$c(z)=\sum_n \frac{c_n}{...
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Why a Majorana spinor is real? [closed]
Two matrices $$\rho^0=\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix} \; \text{and}\; \rho^1=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$$ represent two dimensional Dirac matrices which ...
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Canonical Variables in Dirac Spinor Field Theory [duplicate]
In S.Weinberg [QFT V1][1] section 7.1, in eq (7.1.15) and (7.1.16), he states that in order to be consistent with the previous-derived anti-commutator relation, we should take $\psi_{\text{n}}$ and $\...
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$C$-number ignored in charge conjugation
In Weinberg’s QFT V1, under equation 5.5.58, he says that an anticommutator ($c$-number) can be ignored when we exchange spinors, $\psi$ and $\bar{\psi}$. I cannot fully appreciate why we can ignore ...
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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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Confusion about whether a fermion field and its conjugate as an Grassmann number
I'm confused about what "a Grassmann-odd number" really means and how does it apply to fermions.
In some text, it says that "if $\varepsilon \eta+\eta \varepsilon =0 $, then $\...
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Geometry of anticommutation relations
I am asking this question as a mathematician trying to understand quantum theory, so please forgive my naivety.
Systems satisfying the canonical commutation relations are naturally modeled with ...
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Why does $[Q,P]=i\hbar$ work for fermion? Shouldn't fermion satisfy anticommuting relation?
For hydrogen, we use $[Q,P]=i\hbar$ for electron, which is a fermion. Does it have a deeper reason such as that we're really considering the proton + electron system, which might be of bosonic nature?
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Help with commutator algebra with fermionic operators
I am struggling to understand how the following is true for the fermionic creation/annihilation operators $a^\dagger, a$: $$[a^\dagger a, a]=-a$$
If someone could walk me through the math derivation ...
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Why do we only consider commutators and anticommutators in QFT?
While studying canonical quantization in QFT, I observed that we quantize fields either by a commutation or an anticommutation relation
\begin{equation}
[\phi(x), \phi(y)]_\pm := \phi(x) \phi(y) \pm \...
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Anticommutator Relation of Quantized Fermionic Field and Fermi–Dirac statistics: How are these related?
I'm reading the Wikipedia article about Fermionic field and have some troubles to understand the meaning following phrase:
We impose an anticommutator relation (as opposed to a commutation relation ...
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What is the motivation for the Self-Dual Canonical Anticommutation Relation (CAR) algebra?
What exactly is the motivation for the use of the Self-Dual Canonical Anticommutation Relation (CAR) algebra in the context of infinite lattice systems? Why not remain with the CAR algebra, as both ...
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Commutation of kinetic energy operator with Hamiltonian
I am basically trying to calculate current energy operator $\hat{\mathbf{J}}_E(\mathbf{r})$ by using Heisenberg equation of motion as
$$
-\nabla\cdot \hat{\mathbf{J}}_E(\mathbf{r})=\frac{i}{\hbar}[H,\...
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Contour integral for commutator of fermionic fields
Suppose we have primary fields $A$ and $B$ which have the OPE,
$$A(z) B(w) = \frac{1}{z-w} = -B(z)A(w), \quad |z| > |w|,\tag{1}$$
so they have fermionic statistics. Now I was curious how this would ...
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Dirac spinor field anti-commutation
I am thinking the anti-commutation property of Dirac field! First, note that the equal time anti-commutation relation (from P&S's QFT):
$$\{ \psi_a(\mathbf{x}),\psi_b^{\dagger}(\mathbf{y}) \}=\...
- 1,515
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Adding a surface term to Dirac action modifies the canonical anticommutation relations [duplicate]
I'm dealing with the following issue: when describing a fermionic field, one can use the typical Dirac Lagrangian $$\mathcal{L}=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi,\tag{1}$$ or the more symmetric ...
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Where do arbitrary phases of wavefunctions go under second-quantization?
As far as I understand, a second-quantized operator in QFT or condensed matter represents a many-body wavefunction (symmetrized for bosons or antisymmetrized for fermions). But every wavefunction is ...
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Can the operator field Dirac equation be expressed as Heisenberg's equation?
The Dirac equation of the operator spinor field is:
$$(i\gamma ^{\mu}\partial _{\mu} -m)\psi =0$$
where $\psi$ is interpreted to be a quantum field.
I'm wondering, can this be derived from the ...
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Is there a Stone-von-Neumann theorem-like result for the canonical anti-commutation relations (CAR)?
The canonical commutation relation (CCR)
$$[\phi(x), \pi(y)] = i\hbar\delta(x-y)$$
is kind of the key to basically any bosonic quantum theory. This is due to many different remarkable properties: By ...
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Anticommutation and Bogoliubove transformation
I am given the following transformation:
\begin{equation}
\begin{bmatrix}
...
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