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.

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What is the meaning of gauge theory and Yang-Mills theory? [duplicate]

I would appreciate it if you guys would help me to understand the idea behind these two concepts: Gauge field and Yang-Mills theory. What I think I understand is: Suppose we have a Lagrangian that ...
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How would we integrating high energy part of a relativistic quantum field theory to get a non-relativistic theory?

Relativistic quantum field theory (RQFT) is a after spring of quantum theory and special relativity. A novel thing in RQFT is the existence of anti-particle. That is, consider a relativistic field $\...
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Charge conjugation and Parity for QCD $\theta$ term

I am studying Charge Conjugation ($C$) symmetry and Parity symmetry ($P$) on QCD $\theta$ term: $\tilde{G}G= \frac{1}{2}\epsilon^{\mu \nu \lambda \rho}G^a_{\mu \nu}G^a_{\lambda \rho} \propto G^a_{0i}G^...
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Can Chiral symmetry violating term in lagrangian violate charge conversation?

The regular Lagrangian is $\mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi$ If we add a chiral violating term $\mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu-me^{i\theta\gamma^5})\psi$ For the ...
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Dynamical critical exponent in stochastic vector model

For stochastic $O(N)$ model given by: $$S[\psi,\phi]=\int \frac{d\omega\, dk^D}{(2\pi)^{D+1}} \left( \vec{\psi}(-k,-\omega).\vec{\phi}(k,\omega) \left(-i\omega + \gamma k^2+r \right) -2T\vec{\psi}(-k,-...
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Deriving equation describing fermion-antifermion field

We know the Lagrangian of massless interacting Dirac field $\mathcal{L}=\bar{\psi}i\gamma^\mu(\partial_\mu-iA_\mu)\psi$ Now consider charge conjugation operator $C=i\gamma^2$ The Lagrangian for charge ...
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1 answer
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Noether charge on complex scalar field

For complex scalar field, we write the Lagrangian as: $$ \mathcal{L}=\partial_{\mu}\phi^{*}\partial^{\mu}\phi-m^2 \phi^{*}\phi $$ with the $U(1)$ symmetry, and under infinitesimal transformation: $$ \...
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How is the Feynman propagator (Green's function) connected with the field?

Let's take a look at the Feynman propagator for a massive scalar field: $$D_F(x-y)=\int\frac{dp^3}{(2\pi)^3}\int\frac{dp^0}{2\pi}\frac{ie^{-ip \cdot (x-y)}}{p^2-m^2}$$ We can use this as the Green's ...
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Classical Mechanics Lagrangian from Underlying Quantum Field Theory

Does the K - T classical mechanics Lagrangian emerge from some structure of the Lagrangian of the underlying QFT?
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Doubt on: $G = SU(2)_{L} \times U(1)_{Y}$ representations, the Chiral Spinor bundle and the "split" of covariant derivative for $G$

Firstly, I've made two other questions $[1]$,$[2]$ concerning the same situation, but I think that this one will clarify better what I'm trying to understand. I'm following the text book $[3]$ and I ...
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Conjugate momentum for constant scalar field

I am reading Witten's Why Does Quantum Field Theory in Curved Spacetime Make Sense?, and I am caught up on what appears to be a straightforward computation. The discussion (on page six) centers around ...
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Deriving Euler-Lagrange equation for vector field in curved spacetime

I'm trying to derive covariant Euler-Lagrange equations for a vector field. The variation of the action should be \begin{gather*} \delta S = \int \text{d}^n{x} \sqrt{|g|} \left( \delta\phi^\mu \frac{\...
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What are the beta functions for electroweak and strong constants of interactions?

As the title says I want to find beta function for electroweak and strong constants ($g$ for W-boson, $g'$ for B-boson and $g_s$ for gluons) Beta function is the function that describes change in ...
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Non-minimally coupled inflation — expansion

In the Wikipedia article on "Inflaton" there appears the following formula: $$S=\int d^{4}x \sqrt{-g} \left[\frac{1}{2}m^2_{P}R-\frac{1}{2}\partial^\mu\Phi\partial_{ \mu }\Phi-V(\Phi)-\frac{ ...
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Pauli-Lubanski vector for Maxwell's equation

In the book quantum field theory by Itzykson and Zuber, page 53, the authors prove that Dirac's equation has spin 1/2 by showing that if $\psi$ is a solution to Dirac's equation, then compute that $\...
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1 answer
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Field shift in free Klein-Gordon theory

I am reading Peskin & Schroeder Ch9 and am stuck on a calculation going from equation 9.36. The problem is essentially a change of variable of a Klein-Gordon field. Beginning, we have an integral ...
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Chern-Simons forms: interpretation and generalizations

Studying again differential geometry, anomalies and topology, I wondered if there is ANY physical interpretations (in terms of QFT or even classical field theory) of the Chern-Simons forms, via vacuum,...
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Why Electron Quantum Field Wants Little Energy But Photon Field Doesn't

In this Quora post: https://qr.ae/pv5tac, it states that the electron quantum field "wants" to reduce the energy it has, so when a particle and an anti-particle interact and the charges ...
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2 answers
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What's the difference between these two lagrangian? [closed]

The lagrangian for scalar field is defined as, $$L=L(\phi,\partial_{\mu}\phi,\partial_{\mu}\partial_{\nu}\phi)\tag{1}$$ $but$ there is also another lagrangian which is defined as, $$L=L(\phi,\nabla_{\...
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Help with an integral in Peskin & Schroeder - QFT

In chapter 2, page 27, eq. 2.51, P&S solves the following integral - $$ \frac{4\pi}{8\pi^3} \int _0 ^\infty dp \ \frac{p^2 \ \ \ e ^{-it\sqrt{p^2 + m^2}}}{2\sqrt{p^2 + m^2}}.\tag{2.51}$$ My ...
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1 answer
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How to properly get low energy effective field theory of superfluid?

I am following chapter 3 of X. G. Wen's book "Quantum Field Theory of Many-Body Systems". The following action for a weakly interacting Bose gas is derived: $$S[\varphi,\varphi^*] = \int dt \...
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1 answer
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How do you multiply a 4x1 spinor by a $SU(3)$ matrix?

I understand multiplication by a complex number, i.e. $SU(1)$ I maybe understand multiplication of the spinor by a 2x2 matrix i.e. $SU(2)$. We probably copy the 2x2 matrix twice to form a 4x4 matrix ...
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Hamiltonian and Lagrangian for a particle on a ring [duplicate]

In the book Condensed Matter Field Theory (A. Altland & B. Simons)(page 498, 2nd edition) they suggest the following Hamiltonian and Lagrangian for a particle on a ring in the presence of a ...
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1 vote
1 answer
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Equal-time Canonical Commutation Relation for a scalar field

In chapter 2 of Quantum Field Theory and the Standard Model, Schwartz derives the equal-time commutation relations of the second-quantised field. Using $$ \phi(\vec{x}) = \int \frac{d^3p}{(2\pi)^3} \...
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1 vote
1 answer
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Why does the Lagrangian have $O(4)$ symmetry after Wick rotating (previously Lorentz symmetry)?

Pertaining to the answer within link. Why is it the case, that for Lorentz invariant Lagrangian $\mathcal{L}$, after Wick rotation, the $O(4)$ invariance is established, thus manifesting itself as ...
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2 votes
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Canonically quantizing the charged scalar field with massive gauge boson

I have a specific confusion in canonically quantizing the theory of a complex scalar field $\Phi$ and a real vector field $V^\mu$, with a Lagrangian density: \begin{align*} \mathcal{L} = -(D_\mu \...
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1 vote
1 answer
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What is the problem with classical fermionic field?

Consider classical fermionic field. We have it's action, equations of motion and so we can get it's solutions, right? For example, we can consider gravitational solutions with fermions (in particular, ...
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Calculating conjugate momenta for a spin-2 field

Consider a symmetric spin-2 field $h_{\mu \nu}$. I have the following Lagrangian for this field: $$\mathcal{L} = - \frac{1}{4}\left(\partial_{\lambda}h_{\mu \nu} \text{ } \partial_{\phi}h_{\alpha \...
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Scalar QED Lagrangian

I get this as my Lagrangian for Scalar quantum electrodynamics: \begin{align*} \mathcal{L} & = -(D_\mu \Phi)^{\dagger}(D^\mu \Phi) - m^2 \Phi^{\dagger}\Phi - \lambda (\Phi^{\dagger} \Phi)^2 - \...
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1 vote
1 answer
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A wind tunnel and 2 strong magnets in the wind tunnel creating a very strong field, how would the wind & magnetic force interact?

Sorry if I get some terminologies wrong I am not a physics major :) If I had a big wind tunnel on earth blowing wind through a strong magnetic field (so the opposite poles of 2 giant magnets creating ...
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Expression of a Lagrangian in other form

I'm reading Matthew D. Schwartz, Quantum field theory and standard model and some question arises In his book, p.133, he says that Any vector field can be written as $$ A_{\mu}(x) = A^{T}_{\mu}(x)+ \...
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Doubt on transformation laws of tensors and spinors using standard tensor calculus and group theory

1) Introduction From standard tensor calculus, here restricted to Minkowski spacetime, we learned that: A scalar field is a object that transforms as: $$\phi'(x^{\mu'}) = \phi(x^{\mu})\tag{1}$$ A ...
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3 votes
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Weinberg's Normal Ordering

I have the exact same question about Weinberg's volume 1 that was posted in physicsforums.com 10 years ago but it was never answered. I would greatly appreciate it if someone knows whether there is an ...
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1 answer
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Proca equation gauge conditions

In massive case without any gauge conditions proca equation can be written as $\partial_\nu(\partial^\nu A^\mu- \partial^\mu A^\nu)+\left(\frac{mc}{\hbar}\right)^2 A^\mu=0$ Since $A_\mu$ is a $n$-...
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3 votes
1 answer
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How fundamental Physics is constructed? [duplicate]

I was wondering, how do theoretical physicist arrive to such fundamental things like Lagrangians or actions. For example, the QED, action is given by: $$ \mathcal S_{QED} = \int_{\mathcal M} {\mathrm ...
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Derivation symmetrization canonical energy EM tensor

The Lagrangian for the electromagnetic field is $\mathcal{L}=-\frac{1}{4} F_{\mu\nu} F^{\mu \nu}$. Noether's theorem yields the following canonical stress-energy tensor: \begin{equation} T^{\mu \nu} = ...
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Why is a photon's energy described by $E^2=(pc)^2$ if a photon field is described by $A^\mu$?

We can use Einstein's famous energy equation: $$E^2=(mc^2)^2+(pc)^2 \tag{1}$$ To find the relativistic energy. I can turn this into quantum mechanics and I'll have the Klein-Gordon Equation: $$\...
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How do take non-relativistic limit of this Lagrangian to obtain the Hamiltonian?

In this paper, it is claimed in equation (1) and (2) that when we take non-relativistic limit, the following Lagrangian (the interaction part) $$L=g \partial_{\mu} a \bar{\psi} \gamma^{\mu}\gamma^5\...
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1 vote
2 answers
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Is it possible to elevate the electric-magnetic duality discrete symmetry to a continuous one?

I'm familiar with Electric-Magnetic duality, where in the absence of source fields one can exchange the $F_{\mu \nu}$ field with the dual field: $\tilde{F}_{\mu \nu}={\epsilon}_{\mu \nu \alpha \beta} ...
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1 answer
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Why is the hamiltonian density written in terms of $\phi$ and $\pi$ only? [duplicate]

Why is the hamiltonian density defined as: $$\mathcal{H}=\dot{\phi}\pi-\mathcal{L}$$ Where $\pi \equiv \frac{d\mathcal{L}}{d\dot{\phi}}$ and $\mathcal{L}(\dot\phi, \nabla \phi, \phi)$ is a function of ...
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4 votes
1 answer
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Domain of definition of a Lagrangian in classical field theory

In classical field theory one has the action: $$S[\phi] = \int_{t_{0}}^{t_{1}}\int_{\Omega}\mathcal{L}(t,x,\phi(t,x),\dot{\phi}(t,x),\nabla\phi(t,x))dxdt$$ and we want to obtain the Euler-Lagrange ...
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Equipartition theorem for continous medium

The equipartition theorem states that if $x_i$ is a canonical variable (either position or momentum), then $$\left\langle x_i \frac{\partial \mathcal{H}}{\partial x_j}\right\rangle = \delta_{ij}\ k T.$...
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2 votes
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Can Lagrangians model all possible dynamics? [duplicate]

We use Lagrangians and variational calculus for almost all of physics, from Newtonian mechanics to QFT. Is there any theorem in mathematics that guarantees that all possible dynamics of objects (say ...
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2 votes
1 answer
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Parity-Odd Term in the Lagrangian

I'm currently reading chapter 94 of Srednicki's book, where he calculates the pion contribution to the nEDM (neutron electric dipole moment), after Eq. 94.22 he writes the following I'm having a hard ...
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4 votes
2 answers
843 views

Are mediums fields?

I understand that, at the more fundamental physical level, waves are phenomena of Fields. Like electromagnetic waves of the electromagnetic field. However, I also know that we can have waves in ...
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1 vote
1 answer
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How can we know the maximum of a scalar field after Lorentz transforamtion? [closed]

Support that we have a field $\phi(x)$. we do the Lorentz transformation for it, namely, $$ \phi(x) \to \phi'(x)= \phi(\Lambda^{-1}x). $$ If the field $\phi(x)$ takes the maximum at point $x=a$, where ...
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Non-minimally coupled inflation

In Wikipedia you can read under the Keyword Inflaton , the Formula: What do the individual formula symbols mean in the following formula: $$S=\int d^{4}x \sqrt{-g} \left[\frac{1}{2}m^2_{P}R-\frac{1}{2}...
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4 votes
1 answer
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Different interpretations of (bundle-theoretic) gauge transformations

The physical minimal coupling procedure is usually expressed mathematically in the language of fibre bundles where instead of local presentations we deal with global objects - gauge fields are ...
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(Giamarchi) How do derivatives arise when bosonizing a 1D fermionic system?

I am reading the second chapter in Giamarchi's "Quantum Physics in One Dimension" to understand the bosonization technique. I have a question about how the following harmonic form for the ...
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About longitudinal - transversal decomposition of vector fields

When analyzing the stability of vector fields, a longitudinal+transversal decomposition is usually performed. Such a decomposition looks like: $$A_\mu = B_\mu + \partial_\mu \phi \tag{1}$$ where $B_\...
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