All Questions
Tagged with quantum-field-theory operators
715
questions
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122
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On which bundle do QFT fields live?
In QFT, there is a vector field of electromagnetism, usually notated by $A$, which transforms as a 1-form under coordinate changes. Since quantum fields are operator-valued, I thought it is a section ...
- 61
2
votes
2
answers
81
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How does inserting an operator in the path integral change the equation of motion?
I am reading this review paper "Introduction to Generalized Global Symmetries in QFT and Particle Physics". In equation (2.43)-(2.47), the paper tried to prove that when
$$U_g(\Sigma_2)=\exp\...
- 133
3
votes
1
answer
249
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Derivation of two-body Coulomb interaction in momentum space
$\newcommand{\vec}{\mathbf}$
In Condensed Matter Field Theory by Altland and Simons, they claim the two-body Coulomb interaction for the nearly-free electron model for a $d$-dimensional cube with side ...
0
votes
0
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31
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Why the Slavnov operator is self-adjoint? [duplicate]
In the context of BRST we can define the Slavnov operator $\Delta_{BRST}$ which generates BRST transformations. My lecture notes claim that $\Delta_{BRST}$ is self-adjoint, but I don't see why.
- 357
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votes
2
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116
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Do different bases of Fock space commute?
$\newcommand\dag\dagger$
Suppose we have a Fock space $\mathcal{F}$ with two different bases of creation and annihilation operators $\{a_\lambda, a^\dag_\lambda\}$ and $\{a_{\tilde \lambda}, a^\dag_{\...
3
votes
0
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106
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The commutation relations of photon and gluon?
In QED, the photon field has the following commutation relations:
\begin{equation}
[A^{\mu}(t,\vec{x}),A^{\nu}(t,\vec{y})]=0, \tag{1}
\end{equation}
where $A^{\mu}(t,\vec{x})$ is the photon filed. ...
- 85
3
votes
1
answer
287
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Time-evolution operator in QFT
I am self studying QFT on the book "A modern introduction to quantum field theory" by Maggiore and I am reading the chapter about the Dyson series (chapter 5.3).
It states the following ...
- 613
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How to get $ H=\int\widetilde{dk} \ \omega a^\dagger(\mathbf{k})a(\mathbf{k})+(\mathcal{E}_0-\Omega_0)V $ in Srednicki 3.30 equation?
We have integration is
\begin{align*}
H =-\Omega_0V+\frac12\int\widetilde{dk} \ \omega\Big(a^\dagger(\mathbf{k})a(\mathbf{k})+a(\mathbf{k})a^\dagger(\mathbf{k})\Big)\tag{3.26}
\end{align*}
where
\...
- 37
4
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0
answers
107
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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).$$
Is this equation satisfied by ...
- 2,226
3
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Application of Callias operator in physics
In his article "Axial Anomalies and Index Theorems on Open Spaces" C.Callias shows how the index of the Callias-type operator on $R^{n}$ can be used to study properties of fermions in the ...
- 31
2
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1
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88
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Why does the mass term not violate particle number conservation in a free theory?
The Lagrangian of a free real scalar field theory is
$$ \mathcal{L} = \frac{1}{2} \partial_{\mu} \phi\; \partial^{\mu} \phi \; - \frac{1}{2} m^2 \phi^2. $$
If we decompose $\phi$ in terms of the ...
- 61
3
votes
1
answer
52
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Deriving OPE between vertex operator: Di Francesco Conformal Field Theory equation 6.65
How does one get Di Francesco Conformal Field Theory equation 6.65:
$$ V_\alpha(z,\bar{z})V_\beta(w,\bar{w}) \sim |z-w|^{\frac{2\alpha\beta}{4\pi g}} V_{\alpha+\beta}(w,\bar{w})+\ldots~?\tag{6.65}$$
...
- 327
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64
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Renormalization of the composite operator $\exp(\phi(x))$
I'd like to calculate $\langle\Omega|\exp(\phi(x))|\Omega\rangle$ for quartic scalar field theory (where $|\Omega\rangle$ is the interacting vacuum) and then renormalize to first order in the coupling ...
- 51
2
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2
answers
148
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Particle Creation by a Classical Source (on-shell mass momenta)
It is noted in Peskin and Schroeder's QFT text that the momenta used in the evaluation of the field operator $\phi(x)$ are "on mass-shell": $p^2=m^2$. Specifically, this is in relation to ...
- 4,399
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1
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76
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Why does a normal ordered product of operators (in CFT) have 0 expectation value?
Why does a normal ordered product of operators (in CFT) have 0 expectation value?
The definition (Francesco - Conformal field theory pg. 174) of the normal ordered product of two operators $A(z)$ $B(z)...
- 327
2
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0
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65
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Calculating LSZ reduction for higher order in fields terms
Consider a theory with only a single massless scalar field $\phi(x)$ and a current $J^\mu(x)$ which can be polynomially expanded as fields and their derivatives and spacetime
\begin{align}
J^\mu(x) = ...
- 78
2
votes
1
answer
93
views
Equivalent definitions of Wick ordering
Let $\phi$ denote a field consisting of creation and annihilation operators. In physics, the Wick ordering of $\phi$, denoted $:\phi:$, is defined so that all creation are to the left of all ...
- 3,350
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57
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Is there any difference between Wick time order and Dyson time order?
Reading A Guide to Feynman Diagrams in the Many-Body Problem by R. Mattuck, I am getting the feeling that I missed something subtle related to time order. When deriving the Dyson series for the ...
- 5,578
3
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1
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129
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What are single-, double- and multi-trace operators in AdS/CFT?
Can someone explain what are single-, double- and multi-trace operators are in AdS/CFT? I am a senior undergrad and only recently started studying AdS/CFT from TASI lectures and could not make much ...
- 904
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answers
44
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Picture Number in String Vertex Operator
How can I know what is the Picture of a particular vertex operator?
For example in 8.3.15 in Polchinski's book Vol.1, the Vertex Operators for the Enhanced Gauge symmetry are given by
\begin{equation}...
- 11
1
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1
answer
54
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Choice of spacetime foliation while quantising a conformal field theory
I was reading Rychkov's EPFL lectures on $D\geq 3$ CFT (along with these set of TASI lectures) and in chapter 3, he starts discussing radial quantisation and OPE (operator product expansion). I ...
- 904
4
votes
1
answer
121
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How do we know at the operator-level that the tadpole $\langle\Omega|\phi(x)|\Omega\rangle=0$ vanishes in scalar $\phi^4$ theory?
I'm a mathematician slowly trying to teach myself quantum field theory. To test my understanding, I'm trying to tell myself the whole story from a Lagrangian to scattering amplitudes for scalar $\phi^...
- 421
2
votes
2
answers
124
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Dictionary between interpretations of field operators
For now, let $\hat{\phi}(x)$ be a quantization of a classical, real scalar field $\phi(x)$.
My understanding is that, for fixed $x$, there are three ways to regard the operator $\hat{\phi}(x)$:
The ...
1
vote
1
answer
55
views
Total momentum operator of the Klein-Gordon field (before limit to the continuum)
I'm following K. Huang's QFT: From Operators to Path Integrals book. In the second chapter, he introduces the Klein-Gordon equation (KGE), and its scalar field $\phi(x)$, which satisfies this equation....
- 11
0
votes
2
answers
88
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Ladder operators and creation & annihilation operators - different between $a$, $b$ and $c$ [closed]
Usually, the ladder operator denoted by $a$ and $a^\dagger$. In some case, people talk about the creation operator and denote it by $c$ and $c^\dagger$. Recently I see another notation, $b$ and $b^\...
- 23
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113
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Confused about square of time-reversal operator $T$
I am reading An Introduction to Quantum Field Theory by Peskin & Schroeder, and I am confused about what is the square $T^2$ of time reversal operator $T$.
My guess is that for $P^2$, $C^2$ and $T^...
- 613
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43
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Conjugate momenta in Radial Quantization
When we radially quantize a conformal field theory, is there at least formally a notion of a conjugate momentum $\Pi$ to the primary fields $O$ which would satisfy an equal radius commutation relation ...
- 482
2
votes
1
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88
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Why reasonable observables are made of an even number of fermion fields?
On Michele Maggiore book on QFT (page 91) is stated, out of nothing, that "observables are made of an even number of fermionic operator" and similar sentences is in Peskin book (page 56).
Is ...
- 613
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4
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623
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How can a QFT field act on particle states in Fock space?
Recently I asked a question that was considered a duplicate. However I felt that the related question didn't answer my doubts. After a bit of pondering I have realized the core of my discomfort with ...
- 4,577
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67
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Is integral of energy-momentum tensor in QFT over a region $R$ self-adjoint?
Consider a quantum field theory in flat 1+1D spacetime for simplicity. Let $T_{\mu\nu}$ be the conserved symmetric stress tensor. One writes operators by integrating the tensor over the whole space, ...
- 184
1
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1
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65
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Parity operator action on quantized Dirac field
I am stuck on equation 3.124 on p.65 in Peskin and Schroeder quantum field theory book.
There they are claiming that:
$$P\psi(x)P=\displaystyle\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\bf p}}}\...
- 413
2
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0
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37
views
Double Discontinuity In CFT
In the paper Analyticity in Spin in Conformal Theories Simon defines the double discontinuity as the commutator squared in (2.15):
$$\text{dDisc}\mathcal{G}\left(\rho,\overline{\rho}\right)=\left\...
- 194
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3
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154
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What does the state $a_k a_l^\dagger|0\rangle$ represent?
Consider the action of the operator $a_k a_l^\dagger$ on the vacuum state $$|{\rm vac}\rangle\equiv |0,0,\ldots,0\rangle,$$ the action of $a_l^\dagger$ surely creates one particle in the $l$th state. ...
- 11.8k
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1
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106
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Does every field correspond to a particle?
I know that particles in QFT are just excitations of its corresponding field. But is it possible to have a field which cannot generate particles?
If yes, what terms must be added to the Lagrangian so ...
- 1,611
2
votes
1
answer
130
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Non-perturbative matrix element calculation
Following Peskin & Schroeder's Sec.7's notation, I would like to compute the matrix element
$$
\left<\lambda_\vec{p}| \phi(x)^2 |\Omega\right>\tag{1}
$$
where $\langle\lambda_{\vec{p}}|$ is ...
- 78
0
votes
0
answers
53
views
What is the allowed operator in a global/ local theory?
While I'm reading Hong Liu's notes, it says:
Now we have introduced two theories:
(a)$$\mathcal{L}=-\frac{1}{g^2}Tr[\frac{1}{2}(\partial \Phi )^2+\frac{1}{4}\Phi^4]$$
(b)$$\mathcal{L}=\frac{1}{g^2_{...
- 368
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0
answers
75
views
What does a quantized field in QFT do? [duplicate]
I'm studying for an exam called Introduction to QFT. One of the main topics in this class is the quantized free fields.
I can now find the fields that solve the Klein-Gordon equation and the Dirac ...
- 19
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52
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On discretization in QFT and second quantization
Some time ago i saw in a QFT lecture series by the IFT UNESP that in QFT we need to discretize space by dividing it into tiny boxes of an arbitrary Volume $ \Delta V $ and then define canonical ...
- 309
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0
answers
64
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Naive approach to a path-ordered functional
For analytic functions, we know that
$$ \langle q'|F(\hat{q})|q\rangle = F[q]\,\langle q'|q\rangle\tag{1} $$
Now, suppose that $q$ depends on $\tau$, promote $F[\hat{q}]$ to a functional, and ...
- 266
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Lorentz invariance (LI) of time ordering operation
At Srednicki after eq. (4.10), we have a discussion about that the time ordering operation. Have to be frame inv. I.e it has to be LI.
He wrote that for timelike separation we don't have to worry ...
1
vote
1
answer
94
views
Why does we quantize fields $\phi(t,x)$ and not $\phi$?
In classical mechanics, the action of a theory is determined by its Lagrangian:
$$S(q) := \int L(q(t),\dot{q}(t),t)dt $$
In the following, let us assume that $L$ does not depend explicitly on time. ...
- 1,131
-3
votes
2
answers
107
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Multi-particle Hamiltonian for the free Klein-Gordon field
The text I am reading (Peskin and Schroeder) gives the Hamiltonian for the free Klein-Gordon field as:
$$H=\int {d^3 p\over (2\pi)^3}\; E_p\; a^{\dagger}_{\vec p}a_{\vec p}$$
This does not seem to be ...
- 4,399
-2
votes
1
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74
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On creation annihilation operators of the free Klein-Gordon field [closed]
I want to calculate multiparticle states like $|\vec p,\vec p\rangle$ from $|0\rangle$. It seems that I would need to compute from things like: $a^{\dagger}_{\vec p}a^{\dagger}_{\vec p}|0\rangle$?
It ...
- 4,399
0
votes
0
answers
63
views
Questions about computing the commutator of the Lorentz generator
I am computing the commutator of the Lorentz generators, from the Eqn (3.16) to Eqn (3.17) in Peskin & Schroeder.
$$
\begin{aligned}
J^{\mu\nu} &= i(x^\mu \partial^\nu - x^\nu \partial^\mu ) &...
- 41
2
votes
2
answers
132
views
Commutator of conjugate momentum and field for complex field QFT
In Peskin & Schroeder's Introduction to QFT problem 2.2a), we are asked to find the equations of motion of the complex scalar field starting from the Lagrangian density. I want to show that:
$$i\...
- 538
1
vote
0
answers
76
views
What is the meaning of twist in OPE?
In Operator Product Expansion (such as explained in Peaking) there appear a quantity for an operator called twist, defined to be $d-s$ where $d$ is the scaling dimension of the operator and $s$ is it'...
- 41
1
vote
1
answer
74
views
Quantization of a massless scalar
Let $t$:time, $r$:distance, and $u=t-r$.
Since any massless particle should propagate along u=const. , we need to change the asymptotic infinity of a massless scalar from time infinity to null ...
- 11
1
vote
1
answer
81
views
Non-Abelian anomaly: why does non-Hermitian operator have complete basis of eigenvectors?
In section 13.3 of his book [1], Nakahara computes the non-Abelian anomaly for a chiral Weyl fermion coupled to a gauge field by making use of an operator
$$
\mathrm{i}\hat{D} = \mathrm{i}\gamma^\mu (\...
- 2,157
3
votes
3
answers
815
views
Transition from position as operator in QM to a label in QFT
In David Tong's lecture "Quantum Field Theory" - Lecture 2, he said that
"In Quantum mechanics, position is the dynamical degree of the particle which get changed into an operator but ...
user385554
2
votes
1
answer
116
views
What's the exact definition of fields in conformal field theory?
For example we work with a 2d scalar field $\phi$. I guess $\phi$, $\partial_z\phi$, $\partial_{\bar z}\phi$ are fields, are there more? Is it true that all fields are in the form of $\partial_z^i\...
- 249