Questions tagged [classical-field-theory]

For questions where the dynamical variables are classical fields, that is, functions of several variables (typically, one time coordinate and several space coordinates). If the question comprises both classical and quantum fields, use the tag [field-theory] instead.

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Current and charges conserved in Klein Gordon equation. What are the links between them?

When we deal with the Klein Gordon Lagrangian, we can find some conserved quantities. For example, we remark that under a space time translation $a^\mu$ we can find that the quantity: $a_\rho T^{\mu \...
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Why is action a functional of $q$ only?

Action of a particle is written as $$S[q]=\int dt\hspace{0.2cm} L(q(t),\dot{q}(t),t).$$ How can I understand why $S$ is a functional of $q$, and not that of $\dot{q}$? Assuming $L=\frac{1}{2}m\dot{q}^...
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Understanding an expression from Tom Bank's book of Modern Quantum Field Theory

In Tom Bank's book Modern Quantum Field Theory he says, Consider a classical machine (an emission source) that has probability amplitude $J_E(x)$ of producing a particle at position $x$ in space-...
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Symmetries of the Hamiltonian of a charged particle in a uniform magnetic field

Consider the Hamiltonian of a charged particle of charge $q$ in a uniform magnetic field $\textbf{B}=B\hat{\textbf{z}}$ is given by $$H=\frac{(\textbf{p}-q\textbf{A})^2}{2m}$$ where $\textbf{p}$ is ...
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Why my 4-divergence term added to a Lagrangian modifies the equation of motion?

I take this Lagrangian: $$\mathcal{L}=\mathcal{L}_0+\partial_\alpha f(\phi, \partial_\mu \phi).$$ In this topic Does a four-divergence extra term in a Lagrangian density matter to the field ...
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345 views

Reconciling the causality of QFT and the Feynman propagator

Unlike the retarded propagator, the Feynman propagator $\Delta_F(x-x^\prime)$ is given by $$\Delta_F(x-x^\prime)=\int\frac{d^3\textbf{p}}{(2\pi)^32E_\textbf{p}}\Theta(t-t^\prime)e^{-ip\cdot(x-x^\prime)...
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232 views

Interpretation of the conserved current in classic Klein-Gordon and Dirac equations

The conserved current in KG is $$j^{\mu}=i(\phi^*\partial^{\mu}\phi-\phi\partial^{\mu}\phi^*) =2p^{\mu}|N|^2$$ where N is a normalization factor. This current can't be understood as a probability ...
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455 views

Classification of field types in QFT

In Classical Field Theory fields are sections of bundles over spacetime. In particular we almost always consider vector bundles. Some examples are: Scalar fields: these are sections of the trivial ...
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Proove that $\frac{d^3k}{(2\pi)^3}\frac{1}{2 k_0}$ is Lorentz invariant [closed]

I would like to check if I understood well the proof of "$\frac{d^3k}{(2\pi)^3}\frac{1}{2 k_0}$ is Lorentz invariant". My question is different from the others linked to this topic because I want to ...
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Klein gordon field and positive/negative energy solutions

I my course we calculated the Klein-Gordon field: $$ \phi(x)= \int \frac{d^3k}{(2 \pi)^3}\frac{1}{2k_0} ~ \left[a(\vec{k})e^{-ik.x}+b^*(\vec{k})e^{i kx}\right]$$ We said that the part $ a(\vec{k})e^{...
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100 views

Derivative of a field for infinitesimal transformation

I have a field : $\phi(x^\mu+\delta x^\mu)$. Where $x^\mu$ is a quadriposition. In my course we wrote : $$ \phi(x^\mu+\delta x^\mu) \approx \phi(x^\mu)+\delta x^\mu \partial_\mu \phi(x^\mu)$$ What ...
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General questions about conservation of the action, Noether theorem in classical field theory

In my course, we considered the following Lagrangian : $$\mathcal{L}(\phi,\phi^*, \partial \phi, \partial \phi^*) = \partial_\mu \phi \partial^\mu \phi^* - m^2 \phi^* \phi $$ We said that we want ...
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Particle creation by a source as explained in A. Zee's QFT in a Nutshell

For the free theory $$W[J]=-\frac{1}{2}\int\int d^4x d^4y J(x)D(x-y)J(y)\tag{1}.$$ Introducing the Fourier transform, $J(x)\equiv \int d^4k e^{+ik\cdot x}\tilde{J}(k)$, we get, $$W[J]=-\frac{1}{2}\int ...
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796 views

Noether charge for Lorentz Transformation in terms of creation / annihilation operators

I am trying to compute the conserved charge for a continuous Lorentz symmetry for a real scalar field in terms of creation / annihilation operators. So I have, $$\mathcal{L} = \frac{1}{2} \partial_{\...
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1answer
60 views

How to properly show that : $\sum_{i} \partial_{i} T_{i \mu}=\partial_0 T_{0 \mu} $ knowing that $\partial_{\alpha} T^{\alpha \mu}=0$

We know that : (*) $\partial_{\alpha} T^{\alpha \mu}=0$ for a field following Klein Gordon equation ($T$ is the energy impulsion tensor). And we say in my QFT course, that because of (*) we have : $\...
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509 views

Infinitesimal transformation of fields

In my QFT course, we are doing some infinitesimal transformations of scalar fields. We do the following : $$ \phi'(x')=\phi'(x+\delta x) =\phi'(x)+\delta x^\mu \partial_\mu \phi(x)$$ But i don't ...
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Hamiltonian for vector quantities in curved spacetimes

How would we define the Hamiltonian of some Lagrangian in a curved background (with a timelike killing vector such that time is defined)? My reasoning would be the following: $$S = \int d^dx \sqrt{-...
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Systems with 'many' conserved quantities

The classical justification for the microcanonical ensemble relies on the fact that most many-body systems have just a 'small' (typically finite) number of conserved quantities (i.e. they violate ...
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1answer
164 views

EOM of the SU(2) gauge fields

I have two complex scalar massive fields realizing the foundamental representation of SU(2) $$ \varphi = \begin{pmatrix} \varphi_1 \\ \varphi_2 \\ \end{pmatrix} ...
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1answer
366 views

Motivation for the Euler-Lagrange equations for fields

In Lagrangian Mechanics it is possible to motivate the Euler-Lagrange equations by means of D'alembert's principle. This is a quite more natural route to follow than to start postulating the least ...
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1answer
291 views

What is the significance of $\hbar$ appearing in classical equation of motion?

Books on QFT treats, any quantum field as quantized classical fields. For example, the Klein-Gordon field is first treated as a classical field $\phi(x)$ obeying classical Euler-Lagrange equation $$\...
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1answer
127 views

Coleman's paper about non-existence of Goldstone bososns in 2D

For a massless scalar field in $D+1$ spacetime dimensions the correlation function is given by $$G(x)=\langle 0|\phi(x)\phi(y)|0\rangle=\int\frac{d^{D+1}k}{(2\pi)^{(D+1)/2}}\frac{e^{ik\cdot (x-y)}}{k^...
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535 views

Why is a nonzero VEV for a spinor field said to break Lorentz invariance?

Consider a spinor field $\psi(x)$. Its vacuum expectation value is given by $$v=\langle 0|\psi(x)|0\rangle.$$ Using the fact that the vaccum is invariant under Lorentz transformation, we get, $$v=\...
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1answer
1k views

How to derive the two Friedmann-Lemaître equations from a Lagrangian?

Consider the Lagrangian of an isotropic-homogeneous spacetime (Robertson-Walker metric), containing a simple scalar field and a cosmological constant (this expression comes from the standard Hilbert-...
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1answer
455 views

Specifying the state of polarization of a photon, a classical polarized beam of light, and connection with $\textbf{E}\pm i\textbf{B}$

In Classical Electrodynamics, the state of polarization of a monochromatic electromagnetic wave is specified by the direction of the electric field. For example, $\textbf{E}=\textbf{E}_0\cos(\textbf{k}...
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1answer
776 views

Relation between Noether's charge and the generator of a $U(1)$ symmetry

Consider a $U(1)$ symmetry of a complex scalar field realized as $$\phi\to\phi^\prime=e^{iQ\theta}\phi.$$ where $Q$ is the generator of the symmetry. The conserved Noether's charge (in $D$-spatial ...
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1answer
788 views

Energy momentum tensor from generalized Noether current

Proceeding like here, let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a ...
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159 views

Boundary condition for solitons in 1+1 dimensions to have finite energy

Suppose a classical field configuration of a real scalar field $\phi(x,t)$, in $1+1$ dimensions, has the energy $$E[\phi]=\int\limits_{-\infty}^{+\infty} dx\, \left[\frac{1}{2}\left(\frac{\partial\phi}...
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Scalar field cosmology model: what should be some “realistic” values?

I'm considering a "classical" model of scalar field cosmology: A simple real scalar field minimally coupled to gravity, with a quartic Higgs-like field potential: \begin{equation}\tag{1} \mathcal{V}(\...
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1answer
250 views

Coupling of current to Maxwell field - when are the equations of motion gauge invariant?

Consider a coupling of a Maxwell field $A$ to some current $4$-vector $j^{\mu}$ through the following term in the action: $A^{\mu} j^{\mu}$. What is needed for this term to leave the equations of ...
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1answer
243 views

Canonical momentum combinations in covariant Hamiltonian formalism

I am interested in a statement of the following paper (arxiv:hep-th/9802115), but I will describe the simplest case. I am interested in a free scalar Lagrangian with mostly plus signature (the above ...
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What is the actual form of Noether current in field theory?

Let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a region $\Omega$ in a $D$-...
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2answers
392 views

Can we define partition function for a classical field theory?

I heard that $d$ dimensional relativistic quantum field theory can be viewed as a $d+1$-dimensional statistical mechanics. Can a relativistic/non-relativistic classical field theory be also looked ...
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403 views

Difference between the idea of renormalization in quantum and classical field theories

In this answer, AccidentalFourierTransform explains that Renormalisation has nothing to do with Classical vs Quantum. Any theory, classical or quantum mechanical, needs renomalisation if and only ...
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1answer
346 views

Renormalization of mass of an electron inside the crystal

In Cheng and Li's book, Gauge theory of Elementary Particle Physics, he essentially says that renormalization has nothing to do with infinities. Even in a totally finite theory, we would still have to ...
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1answer
362 views

How do the quantum fluctuations of Goldstone modes forbid SSB in $1+1$ dimensions at $T=0$?

The Mermin-Coleman-Wagner theorem states that quantum fluctuations of Goldstone modes in d=1+1 dimensions are "strong enough" to destroy the possibility of Spontaneous Symmetry Breaking (SSB) at $T=0$....
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How are the vacua of Yang-Mills theory connected by or changed by instanton effects?

Warning: This question is based on poor/immature understanding of instantons and the vacuum structure of the SU(N) Yang-Mills theory. Different vacua of the SU(2) Yang-Mills theory and the ...
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1answer
846 views

Understanding instantons in pure Yang-Mills theory

Yang-Mills Instantons are defined as finite action solutions to the corresponding Euclidean equation of motion. If I understood it correct, then instantons are those classical gauge field ...
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264 views

Canonical vs. Noether momentum for longitudinal waves on a 1D chain

Consider longitudinal vibrations of particles on a line connected by springs. Setting all constants to one, the Lagrangian is $$L = \frac12 \sum_i \dot{\phi}_i^2 - (\phi_i - \phi_{i-1})^2.$$ Here, $\...
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549 views

Physical interpretation of the generators of the conformal symmetries

The Poincare group has ten generators, which have the physical interpretation of energy, momentum, angular momentum, and the system center of mass, and which are of course conserved in any Poincare ...
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367 views

Is general definition of the stress-energy tensor ambigous up to a constant factor?

The following snippet is from Wald: The factor $\alpha_M$ seems to be unmotivated here, moreover, as Wald clearly shows, the KG-field and the EM-field have different $\alpha_M$s. However, from ...
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244 views

Electric field lines dilemma

I recently learnt the concept that electric field lines do not cut each other this statement was proved with the logic that : " If electric field lines would intersect then there would be ultimately ...
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1answer
366 views

Commutation relations between fields and generators

What is the meaning and significance, for example, of the following commutators $[\phi,P_\mu]=i\partial_\mu\phi$ and $[\psi,J_{\mu\nu}]=(i(x_\mu\partial_\nu-x_\nu\partial_\mu)+\frac{1}{2}\sigma_{\mu\...
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1answer
375 views

How can I derive the most general scalar field Lagrangian from locality?

My understanding of the principle of locality in a field theory demands that field degrees of freedom interact locally. For example, $\phi(x)$ at the spacetime point $\phi(x+\delta x)$ can interact ...
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1answer
439 views

Taylor expansion of a scalar function of a four-vector

Consider a scalar function $\phi(x^\mu)$ of a four-vector $x^\mu=(x^0,x^1,x^2,x^3)=(ct,x,y,z)$. What is the Taylor expansion of $\phi(x^\mu+\delta x^\mu)$ for infinitesinal $\delta x^\mu$? In ...
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1answer
454 views

Why can a Lorentz transformation not take $\Delta x^\mu$ to $-\Delta x^\mu$ when $\Delta x$ is time-like? [duplicate]

In Peskin and Schroder's Quantum field theory text (page 28, the passage below Eq. 2.53), it asserts that for a spacelike interval $(x-y)^2<0$, one can perform a Lorentz transformation taking $(x-y)...
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641 views

Can the finite dimensional irreducible $(j_+,j_-)$ representations of the Lorentz group $SO(3,1)$ be unitary?

Since the Lorentz group $SO(3,1)$ is non-compact, it doesn't have any finite dimensional unitary irreducible representation. Is this theorem really valid? One can take complex linear combinations of ...
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3answers
138 views

Why is position considered a label in classical field theory?

I am currently researching into classical field theories and have come across the idea of a position being considered a label in field theory, rather than a dynamic variable. I am not sure why this ...
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556 views

Is “Field” a more fundamental quantity or “Force”(in classical mechancis)?

Consider an isolated system consisting of two particles. We can say the two particles are exerting gravitational forces to each other due to their masses. Also we can say each particle has a ...
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How does the Hubbard Stratonovich transformation decouple interactions?

I'm having trouble understanding how the Hubbard Stratonovich (HS) transformation decouples equations via the introduction of a field variable. The particular problem I'm facing is a derivation in ...

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