Questions tagged [field-theory]
For questions where the dynamical variables are fields, that is, functions of several variables (typically, one time coordinate and several space coordinates). Comprises both classical field theory and quantum field theory. Use this tag when the question applies to both classical and quantum phenomena. Otherwise, use the specific tag instead.
2,681
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What are non-propagating fields?
I have read at different places that in 3 spacetime dimensions, there are NO propagating gravitational degrees of freedom. This seems to imply that we have only "non-propagating" degrees of ...
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Fourier Transformation if we only have relative coordinates
Let’s say we want to do a Fourier Transformation (FT) of a function $f(t-t‘,r-r‘)$ i.e. the function to be Fourier transformed only depends on relative coordinates. This is for example the case if we ...
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Calculations with co- and contravariant formalism in QFT
i have another question regarding calculations with the co- and contravariant formalism in QFT. It is not that i don't understand all of this, but most of the time i'm missing some "middle" ...
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Integration by parts on generic tensors
I try to rephrase here a my question (https://math.stackexchange.com/q/4661784/), explaining more specifically the case.
Given a lagrangian $L=L(\theta_{\mu\nu},\phi)$ , the conserved Noether current ...
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4
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What is field? A form of matter or just a function of position? [duplicate]
I am new to electromagnetism. A few days ago, I decided to learn about electromagnetism through different books and i came across 2 different definitions about fields:
A “field” is any physical ...
- 161
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Three dimensional classical continuum limit, wave equation
Many textbooks of classical mechanics or classical field theory mention that a three dimensional "string" (the continuum limit of a lattice) leads to/can be described by the 3 dimensional ...
- 41
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Do EM equations indicate that charged bodies are the source of the EM field?
As the title indicates, I'm not sure if Maxwell's equations or Gauss' law indicate that charged bodies are the source of the EM fields.
This is question comes from the reading of this paper.
As Mario ...
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Question on the indexes of the lagrangians describing gauge theories
For a gauge group $SU(3)_{C}$ we can construct its principal and associated bundles; we can introduce spinor fields via spin structures and spinor bundles and so on, arriving in a lagrangian theory ...
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How is the $SU(2)_L$ conjugation applied?
I'm reading a paper where they introduce the lepton doublets $L$ and "their $SU(2)_L$ conjugations" $\tilde{L}$, which I'm guessing means
$$
\tilde{L} = i\sigma_2L^*.
$$
After $\textit{vev}$,...
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Fundamental group of configuration space of gauge theories
If I consider the space of all the gauge fields $A_{\mu}$ (call this $A$) and a proper gauge group $\Omega_*$, I know that the fundamental group $\pi_1(A)=0$ and the for the gauge group, for example $...
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Is the sign of the mass in the Dirac action irrelevant? [duplicate]
In even dimensions all the representations of the gamma matrices are equivalent, in particular $\gamma^\mu$ and $-\gamma^\mu$ are equivalent. Usually the Dirac Lagrangian is
\begin{equation}
\psi^\...
- 116
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Relation between Lorentz transformations in QFT and GR [duplicate]
I often have difficulty expressing certain doubts because I am not (not even my self, yes) fully aware of what's going on that bothers me, so forgive me if the question isn't the clearest.
I noticed ...
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About the Classical Scalar Field Lagrangian in Flat Space FLRW Spacetime
So the action for a scalar field in spacetime is typically given as:
$S[\phi]=\int dx^4 (\frac{1}{2}\partial^\mu\phi \partial_\mu \phi - V(\phi))$, thus
$\mathcal{L}[\phi] = \frac{1}{2}\partial^\mu\...
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Description of a Classical Klein-Gordon Field with Momentum Distribution
So if I want to solve the free KG equation, the solution is of the form
$$\phi(x) = \int_{p\in \mathbb{R}^4}\frac{1}{2\omega_\vec{p}} \left( a(\vec{p})e^{-ipx} +b(\vec{p})e^{ipx} \right),$$
where the ...
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Gauge invariance to solve Dirac equation in external EM field
Consider a Dirac equation in external EM field $A_\mu$
$(i\gamma^\mu\partial_\mu-m)\psi=q\gamma^\mu A_\mu\psi$
Consider a solution without EM field $\psi_0$. Let us do the gauge transformation $\psi_0\...
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Does there exist a square root of Euler-Lagrange equations of a field? (Factorization)
Does there exist a square root of Euler-Lagrange equations $\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0$ in the sense that $(x+...
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Is the lattice spacing $a$ a dangerously irrelevant parameter?
Near a renormalization group fixed point, we can perform a scale transformation of length $L' = b^{-1} L$. In this case the relative lattice spacing should transform as $a' = b^{-1} a$. After $n$ ...
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Interpretation of $\phi^n$ terms in Lagrangian density
Why in QFT are $\phi^n$, where $n>2 $, terms in your lagrangian density interpreted as interaction terms?
so $\phi^4$ is considered a self-interaction term.
Similarly for two different fields $\phi,...
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Taylor expansion in momentum integral
On Ashok Das' book "Finite temperature field theory", page 21, the book introduces the thermal mass correction to scalar field.
$$
\begin{aligned}
\Delta m^2 & =\Delta m_0^2+\Delta m_T^2 ...
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Understanding the Energy-Momentum Tensor for the Klein-Gordon Field
On Peskin & Schroder's QFT Book, page 19, they give us the conserved charge associated with spatial translations (equation 2.19):
$$P^i=\int T^{0i}d^3x=-\int\pi \partial_i\phi d^3x$$
where $T^\mu_{...
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Proof that the axial current is conserved in classical QED
I am trying to use the Lagrangian of QED (without kinetic terms for photons) to prove that the axial current of QED satisfies $\partial_\mu j^\mu_5 = 2im\bar\psi\gamma^5\psi,$ where $j^\mu_5 = \bar\...
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Field representations - problem with differential
So when we talk about scalar fields in spacetime, we set that $\phi(x) \rightarrow \Lambda\phi(x)\equiv\phi'(x'). $ So that in the same event, the field value is the same for different frames $\phi'(...
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A question on QED
In QED it's given a current of the particle described by wave function $\psi$
$$j^\mu=\bar{\psi}\gamma^\mu\psi.$$
If we substitute positive- and negative-energy solutions we get that
$$j_{+}^0=j^0_-,$$...
- 719
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Does the field $\Psi$ have different representation for different generator $T_R^a$?
I'm learning the non-abelian gauge theories. Suppose we have a set of (general) fields $\Psi^\alpha(x)$ transforming in a given representation $R$ of the gauge group, with $\alpha, \beta = 1,..., \dim(...
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Finding equation of motion for given Lagrangian with respect to metric
Given the following action in $d$ dimensional $(0,1,...,d-1)$ curved spacetime:
$$ S= \int d^dx\sqrt{-g}\mathscr{L}[\chi,\Phi,g^{\mu\nu}] $$
Where:
$$\mathscr{L}=e^{-2\Phi} \left(-\frac{1}{2\kappa^2}[...
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2
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Is Dirac theory just a real Clifford algebra?
The gamma matrices $\gamma^\mu$ appearing in the Dirac equation span the Clifford algebra ${\cal Cl}_{1,3}$ over real numbers. They are generators of Clifford algebra in that sense that their products:...
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Relationship between Lagrangians describing a particle interacting with a scalar field
In Susskind's Particles and Fields lecture, he considered the Lagrangian obtained by considering a particle and the effects of a scalar field $\phi(t, x)$ with coupling constant $g$ on the particle (...
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Why is $\bar\phi(-k) = \bar\phi ^*(k)$?
In Peskin and Schroeder chapter 2 p. 20, they claim that for a real field $\phi(x)$, its Fourier transform $\bar{\phi}(k)$ obey
$$\bar{\phi}(-k) = \bar{\phi}^*(k)$$
I am confused as to why this is ...
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Understanding the physics behind representations of the Lorentz group for multiplets
In Peskin & Schroeder they state that:
...if $\Phi_a$ is an $n$ component multiplet, the Lorentz transformation law is given by an $n \times n$ matrix $M(\Lambda)$: $$\Phi_a(x) \rightarrow M_{ab}(...
- 1,768
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Derivatives as "dynamical variables of the theory"?
This question will probably sounds very broad.
So, I work with classical general relativity, a bit of field theory and semiclassical gravity. I have a friend, though, that works with things like ...
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Consistency of Coordinate transformation with Conformal transformation
I have a confusion about consistency of coordinate transformation with conformal transformation (in 2-dimensional spacetime.)
We know that a general tensor like $T_{\mu\nu}$ transforms as:
$T^{'}_{\mu\...
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2
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Can the $\eta_{\mu\nu}\mathcal{L}$ term in canonical energy–momentum tensor be omitted?
From Noether theory we can define the canonical energy–momentum tensor as
\begin{equation}
T_{\mu\nu}\equiv\frac{\partial\mathcal{L}}{\partial(\partial^\mu\phi)}\partial_\nu\phi-\eta_{\mu\nu}\mathcal{...
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Path integral quantization of QED [closed]
I am trying to derive the correlation functions in QED within the path integral formalism. As we know, the QED Lagrangian is given by
\begin{align}
\mathcal{L}_{\mathsf{QED}} = \bar{\psi}(...
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1
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Lorentz invariance of the Dirac equation and implicability of the Klein-Gordon equation from the Dirac equation [closed]
I am reading the Peskin & Schroeder's Introduction to quantum field theory, p.42~43 and don't understand some points. In their book p.42 they say that
"To show that it (the Dirac equation) is ...
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Where does the integral go to in the field decomposition for Klein Gordon Field?
I am following Peskin Schroeder's book on QFT. In Chapter 2, page 20, they propose expanding the Klein Gordon Field in terms of the Fourier modes:
$$\phi (x,t)=\frac{1}{(2\pi)^3}\int d^3p e^{ipx}\phi(...
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Why there is a 3-point interaction in this theory?
I'm studying the Lagrangian
$$
\mathcal{L} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi+\lambda\phi\partial_\mu\phi\partial^\mu\phi
$$
And am trying to work out the Feynman rule for the 3-point ...
- 1,305
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Taylor expansion of some Lagrangian (Understanding the Blundell's Quantum field theory, Example 26.5)
I am reading the Lancaster, Blundell's Quantum field theory for the Gifted Amateur, p.243, Example 26.5 and I can't understand some sentences and I don't know how to expand some Lagrangian.
I am a ...
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Ostrogradsky instability and fractional derivatives
Are fractional derivatives (or even more generally differentegrals) also under the scope of the Ostrogradsky instability theorem?
- 6,171
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Klein-Gordon equation in FRW spacetime
The metric for frw spacetime is $$ds^2=a(n)^2(dn^2 - dx^2)$$ where $dn$ is the conformal time differential form. The Klein Gordon equation in curved spacetime is $$\left(\frac{1}{g^{1/2}}\partial_{\mu}...
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Does every field theory have a susy completion?
I have a limited understanding of supersymmetry. I understand that a theory that has supersymmetry can have the supersymmetry broken, leaving at low energy what looks like an ordinary quantum field ...
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Generalized Stokes theorem in superspace
Do the generalized Stokes theorem apply in superspace? Any issues or uncommon behaviour of the gradient, divergence and rotational in superspace?
- 6,171
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Gauge transformation of an adjoint left-handed Weyl spinor in $\rm SU(2)$ fundamental representation
I have a left-handed Weyl spinor field $\Psi_L$ in the fundamental representation of the $\rm SU(2)$ gauge group, which transforms $\Psi_{L,i} \rightarrow \Psi_{L,i} + i\theta^at_{ij}^a\Psi_{L,j}$.
...
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Problem With Perturbation Theory of Klein-Gordon Field
I have a confusion regarding the perturbative correction to the Klein-Gordon field in the presence of interactions (e.g. a potential).
Consider a theory in which the equation of motion is given by
$$
\...
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Wave operator in Kerr spacetime: change of coordinates
The wave equation for a scalar field, in Kerr geometry and in Boyer-Lindquist coordinates, reads:
$$-\left[\frac{(r^2 + a^2)^2 }{\Delta} - a^2 \sin^2\theta \right] \partial^2_t \Phi - \frac{4Mar}{\...
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Field vs Coordinates Variation (Translation)
Under a translation $x^{\prime \mu}= x^{\mu} + \delta x^{\mu}$, we have that for a scalar field it holds: $\phi^{\prime}(x^{\prime}) = \phi(x)$. (actually in general this is true for any field if the ...
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What is the difference between electromagnetic wave and electromagnetic field? [closed]
I am confused about the difference between electromagnetic waves and electromagnetic fields. Can you please explain the distinction between the two and how they relate to each other?
user356659
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Understanding Example 36.7 in the Blundell's Quantum field theory
I am reading Blundell's Quantum field thoery for the gifted amateur, p.332, Example 36.7 and stuck at understanding some calculation.
In the example, he expresses
$$ \Sigma_{s=1}^{2}u^{s}(p)\bar{u}^{s}...
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Representation of the Majorana Fermion
In Schwartz's QFT, in the Weyl basis, Majorana fermions are written on the same footing as Dirac fermions, as matrix
$$
\psi_{Majorana}=(\psi_L \quad i\sigma_2\psi_L^*)^T
$$
I don't understand the ...
- 175
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1
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Why the classical Euler-Lagrange equation is assumed when deriving the Noether's conserved current?
As known, in QFT, the conserved currents, such as the energy-momentum tensor, can be derived from the Noether's theorem and expressed as the product of the field operators. These conserved currents ...
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Gauge symmetry and Gauge Transforms
In QFT or CFT, say the action is invariant under some local transformation. Can we call that transformation a Gauge transform?
There is a specific notion of gauge transform in math which is defined as ...
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