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4
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1answer
82 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
1answer
62 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
65 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
91 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
122 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 ...
1
vote
0answers
118 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, ...
3
votes
1answer
93 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
94 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 ...
1
vote
2answers
102 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 ...
0
votes
0answers
40 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
88 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
42 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 ...
2
votes
1answer
107 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 ...
3
votes
2answers
140 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 ...
1
vote
0answers
129 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 ...
0
votes
1answer
280 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 ...
0
votes
1answer
57 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 ...
4
votes
6answers
809 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
101 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
1answer
58 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 ...
3
votes
1answer
160 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
107 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 ...
0
votes
2answers
51 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 ...
1
vote
1answer
51 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 ...
1
vote
2answers
629 views

Proving the Lorentz invariance of the Lorentz invariant phase space element

I have been looking around for a satisfactory answer to prove that $$\frac{d^3\vec{p}}{2E_{\vec{p}}}$$ where $E_{\vec{p}}=+\sqrt{(|\vec{p}|c)^2+(mc^2)^2}$, is Lorentz invariant. The standard answer ...
0
votes
1answer
57 views

Notion of distance in a Conformal Field Theory

I'm confused about the how the notion of distance is used in Conformal Field Theory. Let's take for example the Operator Product Expansion (OPE). In a conformal field theory, due to the scale ...
1
vote
2answers
144 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
2answers
102 views

Variation of a term in the Lagrangian

I don't understand why $$\frac{\delta}{\delta\phi}\left(\frac12\partial^\mu\phi\partial_\mu\phi\right)~=~\partial^\mu\partial_\mu\phi.\tag{1}$$ If we use integration by parts, there should be a minus ...
0
votes
1answer
184 views

Hamiltonian density of classical Klein-Gordon field

I am working my way through Peskin and Schroeder section 2.2 and trying to show that $T^{00}$ is equivalent to the expression $\frac{1}{2}\pi^2-\frac{1}{2}(\nabla \phi)^2-\frac{1}{2}m^2\phi^2$ in ...
0
votes
1answer
170 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 $(2+1)$ ...
0
votes
1answer
40 views

Lagrangian density with explicit $x_\mu$ dependence

In the Quantum Field Theory book, by Ryder, he says that a Lagrangian density of a field can also be an explicit function of $x_\mu$ if the field interacts with external sources. Can someone give an ...
3
votes
1answer
61 views

What is it that Lagrangian density with only bilinear terms always corresponds to free field theory?

Is there an intuitive proof of this fact? (Maybe connected in some way to Central Limit Theorem?).
3
votes
1answer
222 views

effect of a simultaneous local and a global $U(1)$ symmetry breaking

EDIT : I am trying to figure out the effect of symmetry breaking in a $U(1)_Y\times U(1)_Z$ invariant lagrangian where $U(1)_Y$ is local symmetry of the Lagrangian and $U(1)_Z$ is a global symmetry of ...
8
votes
1answer
464 views

Noether currents in QFT

I am trying to organize my knowledge of Noether's theorem in QFT. There are several questions I would like to have an answer to. In classical field theory, Noether's theorem states that for each ...
1
vote
0answers
62 views

Noether current scale transform of EM

I'm trying to solve a question about scale tranform of free EM. I got the next trnaform rules (these two line where EDITed later) $\delta x = -bx$ $\delta A = bA$ the current I got $D^\mu = ...
1
vote
1answer
39 views

Why is it that every locally conformal transformation can be extended to a global conformal transformation for D>2?

In D=2, we can have locally analytic transformations that cannot be globally well-defined. However, for CFTs in D>2, we have only the global group. Why is that? Also, is it a statement that depends ...
0
votes
1answer
111 views

permanent magnet energy field [duplicate]

Please explain to me this: What is the physical mechanism of the field and atraction force; in which way a force (electro, magnetic or gravity) is transmited between two objects? I will try to ...
9
votes
1answer
313 views

What is the quantum state of a static electric field?

This is something that I've been curious about for some time. A coherent, monochromatic electromagnetic wave is well described by a coherent state $|\alpha\rangle$. The quantum treatment of the ...
0
votes
1answer
84 views

Why are Majorana fields usually used to introduce gravity in the Rarita-Schwinger Lagrangian?

When first introducing the gravitational interaction for a spin-3/2 Rarita-Schwinger field, Majorana fields are usually used (see for example here at chapter 4, or in Ramond, (6.4.112) ). Why is ...
2
votes
2answers
175 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 ...
0
votes
1answer
54 views

What does Field theory concern? [duplicate]

I wounder to know what is the basic idea of field theory? Field theory wants to concern what aspects, and formulate what problems in nature?
2
votes
1answer
93 views

Why is $\mu_0$ missing in EM formulas in Peskin and Schroeder?

In this post, $\hbar=c=1$ units are used throughout. It is well known that the action of classical electromagnetism is given by $$\mathcal S_{\text{Maxwell}} = \int ...
2
votes
0answers
237 views

Equations of motion in Maxwell-Chern-Simons theory [closed]

I've started with the Maxwell-Chern-Simons lagrangian (in 2+1 dimensions): $$L_{MCS}=-\frac{1}{4}F^{\mu \nu}F_{\mu\nu}+\frac{g}{2} \epsilon^{\mu \nu \rho}A_\mu\partial_\nu A_\rho$$ From this ...
3
votes
2answers
252 views

Classical EM : clear link between gauge symmetry and charge conservation

In the case of classical field theory, Noether's theorem ensures that for a given action $$S=\int \mathrm{d}^dx\,\mathcal{L}(\phi_\mu,\partial_\nu\phi_\mu,x^i)$$ that stays invariant under the ...
2
votes
1answer
103 views

How to derive the conserved current of the Klein Gordon equation?

Similarly to the probability current in non-relativistic quantum mechanics, there is a conserved current for the Klein Gordon equation, however a different one. I'm trying to calculate that. The KG ...
0
votes
2answers
125 views

How can I prove that $\langle\Omega\vert \phi(x) \vert\Omega\rangle \langle\Omega\vert\phi(y)\vert\Omega\rangle=0$ for a scalar field?

From Peskin-Schroeder, p.212: The term $$ \langle \Omega | \phi(x) | \Omega \rangle \langle \Omega |\phi(y) | \Omega \rangle$$ is usually zero by symmetry; for higher-spin fields, it is zero ...
3
votes
1answer
139 views

Calculation of the Abelian Induced Chern-Simons Term

In Gerald Dunne's paper "Aspects of Chern-Simons Theory" (http://arxiv.org/abs/hep-th/9902115) I'm a little confused as to how equation (225) on page 53 is obtained. Equation (225): ...
3
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0answers
101 views

Field Lagrangian <--> Particle Lagrangian

The action-functionals describing the motion $\mathbf{x}:[a,b]\to \mathbb{R}^3$ of a free particle of mass $m$ and the evolution $\varphi:[a,b]\times \Omega\to \mathbb{R}$ of a free scalar field of ...
1
vote
1answer
91 views

Obtaining momentum operator $P^\mu$ from Lagrangian and energy-momentum tensor $T^{\mu\nu}$

I am pretty new to quantum field theory. Given the Lagrangian density, $$ \mathcal{L} = \frac{1}{2} ( \partial_\mu \phi ) ( \partial^\mu \phi ) - \frac{1}{2} m^2 \phi^2 $$ and its energy-momentum ...
3
votes
1answer
109 views

Trying to show that the current is conserved

$ \newcommand{\p}{\partial} $ I am trying to show that the current $J^{\mu} = (\gamma_{\nu}\partial^{\nu} \phi - m\phi)\gamma^{\mu}\psi$ is conserved for all fields that satisfy the Klein-Gordon and ...