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8
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4answers
2k views

Lagrangian to Hamiltonian in Quantum Field Theory

While deriving Hamiltonian from Lagrangian density, we use the formula $$\mathcal{H} ~=~ \pi \dot{\phi} - \mathcal{L}.$$ But since we are considering space and time as parameters, why the formula ...
1
vote
1answer
49 views

If time-like paths are geodesics, what physical principle applies to space-like intervals?

If I have a number of particles interacting with one another locally, then the center of mass of the system moves along a geodesic. Taking this further with the particles interacting via an EM field, ...
3
votes
0answers
45 views

Renormalization of Auxiliary Fields

I have the following non-linear sigma model (the base space $\mathcal{M}$ is Euclidean): $$ \mathcal{L}=\dfrac{1}{2\alpha}\int_{\mathcal{M}}\mathrm{d}^2\sigma\ \partial^2X^{\mu}\partial^2X_{\mu} $$ ...
0
votes
1answer
12 views

Meaning of these terms for stress fields?

I'm from a math & comp sci background and I'm currently looking at facture theory which deals with stress fields. Can someone explain to me what the following terms represent in the context of a ...
2
votes
2answers
85 views
0
votes
1answer
37 views

quantum fluctuations and the virtual particles

In the introduction of chapter-12 of “An Introduction to Quantum Field Theory” by Peskin and Schroeder I encountered this line: “The quantum fluctuatuations at arbitrarily short distances appear in ...
5
votes
1answer
144 views

Functional derivatives as distributions

I have asked this on math stack exchange, due to its mostly mathemtical content, but aside from one upvote and minimal views it has not garnered any attention, so I am trying here as well. This isn't ...
2
votes
1answer
52 views

Poincare non-invariance in real world and field theory

This may be a very blunt question but I wonder why we always use Poincare invariant Lagrangians in field theory. After all, the entire world around us is by no means homogeneous, isotropic and so on. ...
1
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0answers
44 views

Why are the particles called irreps of Poincare group? [duplicate]

Why are particle excitations called irreducible representation of the Poincare group? It will be very helpful if someone can illustrate with one concrete example of a particle. EDIT : But how does ...
0
votes
1answer
103 views

How to obtain the asymptotic behavior of Green's function?

This question arose from Eq.(9.135) and Eq.(9.136) in Fradkin's Field theories of condensed matter physics (2nd Ed.). The author mapped quantum-dimer models to an action of monopole gas in $(2+1)$ ...
12
votes
1answer
720 views

Could this model have soliton solutions?

We consider a theory described by the Lagrangian, $$\mathcal{L}=i\bar{\Psi}\gamma^\mu\partial_\mu\Psi-m\bar{\Psi}\Psi+\frac{1}{2}g(\bar{\Psi}\Psi)^2$$ The corresponding field equations are, ...
6
votes
1answer
177 views

How do creation operators change with time in an interacting theory?

When studying the quantization of a field theory with free fields, the creation operators $a^\dagger(k)$ are independent of time. In an interacting theory, they are time-dependent, and therefore ...
1
vote
1answer
65 views

Equivalence principle for test fields

My question is very simple. We all know that, for a test particle(classical) in a gravitational field, the motion is only determined by the geodesic lines(let's forget about the initial conditions for ...
0
votes
1answer
50 views

Why is the solution of the $\phi^6$ potential not a soliton?

Consider a theory with a $\phi^6$-scalar potential: $$ \mathcal{L} = \frac{1}{2}(\partial_\mu\phi)^2-\phi^2(\phi^2-1)^2. $$ I solved its equation of motion but found that the general form of its ...
2
votes
1answer
82 views

Invariance of Fermionic action under Lorentz transformations

Suppose I have an Lagrangian $$\mathcal{L} = \frac{1}{2}g_{ab} \bar{\psi}^a \Gamma^k \partial_k \psi^b $$ and I want to show it's invariance under the infinitesimal Lorentz transformations $$\delta ...
1
vote
0answers
12 views

What's the shape of electric field line in a rectangular metal under varying magnetic field

We know that,the shape of electric field line in a cylinder under varying magnetic field is circle,and I wonder,what's the situation if it is a piece of metal with rectangular cross section? I think ...
0
votes
3answers
97 views

The Higgs field vs the Higgs boson

As I see it the Higgs boson is the mediating particle of the Higgs field and get its own mass from the Higgs field. Isn´t this circular? I mean, for instance, an electron creates a radial electric ...
0
votes
1answer
193 views

What are the boundary conditions for EM waves normally incident on the interface between two dielectric media?

An EM wave, amplitude $E_0$, frequency $\omega_0$, is incident upon a material with relative permittivity (dielectric function) $$\varepsilon \left( z \right) = \left\{ \begin{gathered}{\varepsilon ...
2
votes
1answer
64 views

How can one prove that there cannot exist a conformal primary, in the case of free field theory, that doesn't saturate the unitarity bound?

In free field theory, the full list of conformal primaries, is given by the Twist-2 operators. These have $\Delta = l+2$, which is also the saturation condition for the unitarity bound for $l \neq 0$. ...
2
votes
1answer
133 views

Noether Current when the Lagrangian depends on second derivative of the fields

Let a Lagrangian density for a field theory of $N$ fields $\left\{\phi_i\right\}_{i=1}^N$ be given. Assume that the Lagrangian density depends on the fields, their spacetime derivatives, and their ...
3
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0answers
61 views

What is the reason for chiral anomalies in condensed matter systems?

If you consider a massless relativistic fermion theory and you perform a chiral transformation, then you realize that while the classical action remains invariant under this transformation the ...
1
vote
2answers
124 views

Does spatial coupling prohibit resonances due to an external source field?

The harmonic oscillator coupled to a sinodial external source $$\frac{\partial^2 x(t)}{\partial t^2}+\omega_0^2 x(t)=F_0\sin(\omega_\text{ext}\ t),$$ has the solution $$x(t)=x(0)\cos(\omega_0 t)+C ...
0
votes
2answers
53 views

what is the result of change in gravitational flux?

As a change in magnetic flux results in induced EMF (electromotive force) likewise what is the result of a change in gravitational flux? UPDATE: Gravitational flux according to me has only ...
4
votes
1answer
50 views

Why should Ward identities only be used with the effective action (as opposed to the generating functional for connected diagrams)?

My question is about the derivation of Ward identities. I will sketch it here in the case of an O(N) symmetric model and point out what it bothering me when I am done. I am being very sloppy with the ...
1
vote
2answers
105 views

Vanishing of conjugate momentum $\Pi^0$ and non-existence of propagator

We know that if we try to quantize the free electromagnetic field without a gauge fixing term added to the Lagrangian, then one of the conjugate momentum density $\Pi^0$ vanishes. We also find that ...
1
vote
1answer
51 views

Sign of matter Lagrangian term in curved space

In field theory the (matter) Lagrangian $\mathcal{L}_m$ is uncertain upto an overall constant multiplying factor (i.e. $\mathcal{L}_m$ and $a\mathcal{L}_m$ yield the same field equation(s) on ...
0
votes
0answers
32 views

Conservation of momentum in field theory

By studying electrodynamics a while now, I came to this question on the conservation of momentum. In Newtonian mechanics the Third Law allows us to prove the conservation of momentum, with the ...
2
votes
1answer
87 views

Hamiltonian field equations constraints

Let's consider the Lagrangian $$\mathcal{L}~=~-\frac{1}{2}(\partial_\mu\phi^\nu)^2+\frac{1}{2}(\partial_\mu\phi^\mu)^2+\frac{1}{2}m^2\phi_\mu \phi^\mu,$$ with Minkowski metric $\eta_{\mu\nu}={\rm ...
2
votes
0answers
60 views

Noether's first and second theorems

My understanding of Noether's first theorem is as follows. Consider a set of infinitesimal transformations that leave the action invariant, that are indexed by $n$ linearly independent parameters, ...
4
votes
1answer
66 views

Noether's theorem in field theory: Jacobian factor

Following my earlier question in this Phys.SE post I have another question regarding the derivation I am struggling through! Considering the variation in the Lagrange density for $x'=x+\delta x$ and ...
2
votes
0answers
43 views

Index notation for a Lagrangian with second derivatives

I'm finding the field equations for a hypothetical Lagrangian with dependence on the second derivative of a scalar field, $L\left(\phi,\phi_{,\mu},\phi_{,\mu\nu}\right)$, and in the analogue to the ...
2
votes
2answers
61 views

Noether's theorem in classical field theory

I am trying to understand the continuum version of Noethers theorem from this source (p 15- 17) however I am stuck on a couple of points. I will go through what I have so far and then ask my questions ...
1
vote
0answers
91 views

Precisely speaking, does photon become massive or the phonon become massive, due to Higgs mechanism in superconductor?

Consider the low-energy field theories of superfluids and superconductors. In superfluids, the spontaneous breaking of the order parameter's phase creates phonons as the massless Goldstone ...
1
vote
1answer
106 views

Difference between a “source dipole” and a “force dipole”

I know quite well what a dipole is and in general what multipole moments are (in the context of, for instance, electrodynamics). What I find myself confused by is something called a "force dipole" in ...
0
votes
0answers
20 views

What are non-local charges?

In integrable systems, for example in the XXX spin chain, one encounters non-local charges (that form a Yangian). They are fine since the Yangian generate an infinite number of them, which gives us ...
0
votes
1answer
57 views

Entanglement entropy in (1+1)d field theory with dynamical critical exponent $z>1$

It was well known that for (1+1)d CFT(z=1) case, we can use the tool of conformal map to derive the formula of entanglement entropy for a finite interval: S ~ $c \log L$. L is the length of the ...
2
votes
1answer
68 views

In general, can a Lagrangian density depend on space-time explicitly?

In an exercise on classical field theories, I'm trying to derive the general formula of the Energy-momentum tensor. According to the formula in the lecture notes, this tensor includes a term of minus ...
2
votes
2answers
152 views

Does Bell's theorem sort out local field theories?

For example the Maxwell's equations is a local theory. It's a set of differential equations that describe how should the state at a point change based on its neighbourhood. Counter example: Newtonian ...
2
votes
1answer
34 views

A question on the functional dependence of the Lagrangian density

I understand that in classical mechanics the state of a particle at a given instant in time is given by its position $q$ and its velocity at that point $\dot{q}$, and given that, for any given point ...
3
votes
2answers
112 views

How to count the number of modes/polarizations of a Gaussian field theory?

A Gaussian (free) field theory is described by a quadratic action of the field, e.g. $S=\int\psi^\dagger K\psi$ (or $S=\frac{1}{2}\int\phi^\intercal K\phi$ for real fields). Usually one just need to ...
0
votes
1answer
68 views

Euclidean classical action

This is the Euclidean classical action $S_{cl}[\phi]=\int d^{4}x\ (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))$. It would be nice if somebody could explain the structure of the potential. I don't ...
4
votes
6answers
706 views

Is electromagnetic vector field a sum of E and B?

I have a hard time to fully understand (classical) electromagnetic field theory with respect to Helmholtz's decomposition. Let me start from Helmholtz's theorem: Any vector field of class ...
0
votes
1answer
37 views

How to prove that the nonlinear completion of free massless spin-2 action must be Einstein-Hilbert action?

There is a saying that the nonlinear completion of free massless spin-2 action in Minkovski spacetime (that is Fierz-Pauli action) must be Einstein-Hilbert action up to Lovelock invariants. I find a ...
0
votes
1answer
72 views

Does the flatness of a gauge field has anything to do with whether it's dynamical?

One common way in studying Symmetry Protected Topological(SPT) phases with a global symmetry G is to promote G to a gauge symmetry and couple the system to a flat gauge field A for G. Then one can ...
0
votes
2answers
36 views

How to define conserved charges in Euclidean field theory?

In a field theory with signature (1,d), conserved charges are obtained by integrating the time component of a conserved current over a spatial region. What are the corresponding equations and ...
0
votes
1answer
48 views

Does invariance under infinite small transformation imply invariance to the finite one?

Let's say that I have finite chiral transform and I would like to show invariance of Dirac's Lagrangian when $m=0$ under it. The chiral transform is defined as: $$\psi(x) \rightarrow \psi'(x) =e^{i ...
2
votes
1answer
60 views

2D square lattice, nearest neighbor and next-nearest connected by springs

For my field theory class I am trying to build the Lagrangian for the following system. Consider a 2D square lattice where the nearest and next-nearest neighbor interactions are modeled by springs ...
1
vote
3answers
89 views

Suppose $\phi(x)$ is a field, how should I interpret $\partial^\mu\phi$ and $\partial_\mu\phi$?

I am really confused by the sub and upperscript notation sometimes. It might be really trivial but I have a difficult time interpreting the following things in for example this Lagrangian ...
8
votes
2answers
341 views

Does Noether's theorem also give rise to quantities conserved over space?

Noether's theorem gives rise to quantities that are conserved over time. But does it also give rise to quantities that are conserved over space?
1
vote
1answer
38 views

Conserved current in a complex relativistic scalar field

For my field theory class I have the following Lagrangian density $$\mathscr{L}=\frac{1}{2}\eta^{\mu\nu}\partial_\mu\phi^*\partial_\nu\phi-\frac{1}{2}m^2\phi^*\phi$$ Where $\eta^{\mu\nu}$ is the ...