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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140
votes
14answers
35k views

What is a field, really?

There was a reason why I constantly failed physics at school and university, and that reason was, apart from the fact I was immensely lazy, that I mentally refused to "believe" more advanced ...
83
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10answers
9k views

Why are differential equations for fields in physics of order two?

What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations? If someone on the street would flat out ask me ...
83
votes
3answers
13k views

Differentiating Propagator, Green's function, Correlation function, etc

For the following quantities respectively, could someone write down the common definitions, their meaning, the field of study in which one would typically find these under their actual name, and most ...
61
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4answers
22k views

Why correlation functions?

While this concept is widely used in physics, it is really puzzling (at least for beginners) that you just have to multiply two functions (or the function by itself) at different values of the ...
55
votes
4answers
12k views

Why treat complex scalar field and its complex conjugate as two different fields?

I am new to QFT, so I may have some of the terminology incorrect. Many QFT books provide an example of deriving equations of motion for various free theories. One example is for a complex scalar ...
44
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7answers
4k views

Why fermions have a first order (Dirac) equation and bosons a second order one?

Is there a deep reason for a fermion to have a first order equation in the derivative while the bosons have a second order one? Does this imply deep theoretical differences (like space phase dimesion ...
38
votes
3answers
6k views

What are quantum fields mathematically?

I'm confused as to how quantum fields are defined mathematically, and I've seen from questions on this site and Wikipedia articles that classical fields are just functions that output a field value ...
37
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4answers
6k views

How do we know that gravity is spacetime and not a field on spacetime?

How do we know that gravity is the curvature of spacetime as opposed to a field, which couples equally to all objects, on spacetime?
37
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1answer
2k views

Can lightning be used to solve NP-complete problems?

I'm a MS/BS computer science guy who is wondering about why lightning can't (or can?) be used to solve NP complete problems efficiently, but I don't understand the physics behind lightning, so I'm ...
37
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1answer
2k views

What, to a physicist, are instantons and the Donaldson invariants?

I study gauge theory from a mathematical perspective. To me, one of the most fundamental ideas is the notion of an instanton on a 4-manifold. To be precise, I have a Riemannian 4-manifold and a ...
35
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4answers
5k views

Is there just one EM field for the whole universe?

Does our universe contain individual magnetic fields ? For example two different magnets, one here on earth and one on mars. Do both of them have their own magnetic field? Or is there only one single ...
33
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5answers
6k views

Why are infinite order Lagrangians called 'non-local'?

And in what sense are they 'non-local'?
29
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2answers
2k views

Can the center of charge and center of mass of an electron differ in quantum mechanics?

Traditionally for a free electron, we presume the expectation of its location (place of the center of mass) and the center of charge at the same place. Although this seemed to be reasonable for a ...
28
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2answers
12k views

Why on-shell vs. off-shell matters?

The definitions between on- and off-shell are given in Wikipedia. Why is it so important in QFT to distinguish these two notions?
25
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1answer
2k views

Why we don't have macroscopic fields of Higgs bosons or gluons?

Why is it that there exists a classical macroscopic field of photons and gravitons but not that of $Z, W^{\pm}$ bosons, gluons or Higgs boson?
24
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5answers
6k views

What is the difference between a complex scalar field and two real scalar fields?

Consider a complex scalar field $\phi$ with the Lagrangian: $$L = \partial_\mu\phi^\dagger\partial^\mu\phi - m^2 \phi^\dagger\phi.$$ Consider also two real scalar fields $\phi_1$ and $\phi_2$ with ...
24
votes
2answers
6k views

Energy-Momentum Tensor in QFT vs. GR

What is the correspondence between the conserved canonical energy-momentum tensor, which is $$ T^{\mu\nu}_{can} := \sum_{i=1}^N\frac{\delta\mathcal{L}_{Matter}}{\delta(\partial_\mu f_i)}\partial^\nu ...
24
votes
2answers
2k views

Wick rotation and spinors

I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; ...
23
votes
1answer
5k views

Noether's Theorem and scale invariance

Noether's theorem usually considers coordinate/field transformations which leave the Lagrangian invariant up to a divergence term, i.e. $$\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$$ ...
22
votes
3answers
1k views

Does 4D ${\cal N} = 3$ supersymmetry exist?

Steven Weinberg's book "The Quantum Theory of Fields", volume 3, page 46 gives the following argument against ${\cal N} = 3$ supersymmetry: "For global ${\cal N} = 4$ supersymmetry there is just ...
21
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4answers
10k 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 $$\...
21
votes
4answers
5k views

Do tachyons move faster than light?

I am trying to understand whether or not tachyons travel faster than light. The linked Wikipedia page shows some seemingly contradictory statements, and they are confusing. For instance, the first ...
21
votes
1answer
2k views

Why does charge conservation due to gauge symmetry only hold on-shell?

While deriving Noether's theorem or the generator(and hence conserved current) for a continuous symmetry, we work modulo the assumption that the field equations hold. Considering the case of gauge ...
20
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2answers
5k views

Physical difference between gauge symmetries and global symmetries

There are plenty of well-answered questions on Physics SE about the mathematical differences between gauge symmetries and global symmetries, such as this question. However I would like to understand ...
20
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2answers
2k views

Relation between Wilsonian renormalization and Counterterm Renormalization

Wilsonian renormalization The answer by Heider in this link points out that when we integrate out high momentum Fourier modes, we end up with Wilsonian effective action (not the 1PI action). This is ...
19
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4answers
2k views

History of Electromagnetic Field Tensor

I'm curious to learn how people discovered that electric and magnetic fields could be nicely put into one simple tensor. It's clear that the tensor provides many beautiful simplifications to the ...
19
votes
1answer
548 views

Is there a field equation which can reduce into all three flavors of spin (zero, one, one half)?

Is there a known particle field equation of a similar form $$ \begin{equation} (\Gamma^n \pi_n)^2 \Psi = (mc)^2 \Psi \tag{1} \end{equation} $$ such that by reducing the number of degrees of freedom ...
19
votes
0answers
543 views

Electric charges on compact four-manifolds

Textbook wisdom in electromagnetism tells you that there is no total electric charge on a compact manifold. For example, consider space-time of the form $\mathbb{R} \times M_3$ where the first factor ...
18
votes
2answers
9k views

What is a non linear $\sigma$ model?

What exactly is a non linear $\sigma$ model? In many books one can view many different types of non linear $\sigma$ models but I don't understand what is the link between all of them and why it is ...
18
votes
2answers
987 views

Can one write down a Hamiltonian in the absence of a Lagrangian?

How can I define the Hamiltonian independent of the Lagrangian? For instance, let's assume that i have a set of field equations that cannot be integrated to an action. Is there any prescription to ...
17
votes
5answers
14k views

Deriving Lagrangian density for electromagnetic field

In considering the (special) relativistic EM field, I understand that assuming a Lagrangian density of the form $$\mathcal{L} =-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{c}j_\mu A^\mu$$ and ...
17
votes
3answers
7k views

Active versus passive transformations

I am a bit confused by the concepts of active and passive transformations. In all the courses I am doing at the moment we do transformations of the form: $$ \phi(x) \rightarrow\phi'(x') = \phi(x) $$ ...
17
votes
2answers
4k views

When is stress-energy tensor defined as variation of action with respect to metric conserved?

In General Relativity Einstein's equation implies that stress-energy tensor on its RHS is conserved (has vanishing divergence), due to the Bianchi identity. Considering variational principles leading ...
17
votes
3answers
3k views

Why can't General Relativity be written in terms of physical variables?

I am aware that the field in General Relativity (the metric, $g_{\mu\nu}$) is not completely physical, as two metrics which are related by a diffeomorphism (~ a change in coordinates) are physically ...
17
votes
3answers
2k views

Global vs. local gauge group in mathematical sense - physics examples?

Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group: Local gauge group $G$. Corresponds to the fibers of the $G$-...
15
votes
5answers
2k views

Form of the Classical EM Lagrangian

So I know that for an electromagnetic field in a vacuum the Lagrangian is $$\mathcal L=-\frac 1 4 F^{\mu\nu} F_{\mu\nu},$$ the standard model tells me this. What I want to know is if there is an ...
15
votes
5answers
3k views

Quantum mechanics as classical field theory

Can we view the normal, non-relativistic quantum mechanics as a classical fields? I know, that one can derive the Schrödinger equation from the Lagrangian density $${\cal L} ~=~ \frac{i\hbar}{2} (\...
15
votes
1answer
348 views

6d Massive Gravity

Massive gravity (with a Fierz-Pauli mass) in 4 dimensions is very well-studied, involving exotic phenomena like the van Dam-Veltman-Zakharov (vDVZ) discontinuity and the Vainshtein effect that all ...
15
votes
4answers
531 views

What is the experimental evidence for the gravitational field having positive energy density?

Recent direct observation of gravitational perturbations attributed to merging black holes and merging neutron stars has reliably confirmed the existence of gravitational waves. The observed fact that ...
14
votes
1answer
4k views

Trick for deriving the stress tensor in any theory

In D. Tong's notes on string theory (pdf) section 4.1.1 he explains a trick for deriving the stress-energy tensor which arises from translations in the base manifold of the field theory (in this case ...
14
votes
1answer
978 views

Is there a systematic way to obtain all conserved quantities of a system?

I'd like to know whether, given a system, there's a way to obtain all the conserved quantities. For instance if the system consists of electric and magnetic fields, the fields must satisfy Maxwell's ...
14
votes
2answers
695 views

Gauge-fixing of an arbitrary field: off-shell & on-shell degrees of freedom

How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?
14
votes
1answer
1k 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 ...
14
votes
2answers
361 views

Why do we need to embed particles into fields?

In QFT we have the so-called embeding of particles into fields. This is discussed at full generality in Weinberg's book, chapter 5. In summary what one does is: From Wigner's classification, for each ...
14
votes
1answer
889 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, $$(i\...
14
votes
0answers
163 views

Is it known what the necessary and sufficient conditions are for the existence of a “3+1 split” (by means of a foliation) of a (Lorentzian) manifold?

When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to ...
13
votes
5answers
2k views

Is energy localised in space?

I may be somewhat confused here, but as I understand it: Energy is a scalar quantity and colloquially objects are said to “possess” say, kinetic energy or potential energy, or rest energy proportional ...
13
votes
4answers
443 views

What makes an equation an 'equation of motion'?

Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint. For example, in the ...
13
votes
5answers
650 views

Why must the field equations be differential?

In Landau–Lifshitz's Course of Theoretical Physics, Vol. 2 (‘Classical Fields Theory’), Ch. IV, § 27, there is an explanation why the field equations should be linear differential equations. It goes ...
13
votes
3answers
361 views

Why is imposing a symmetry on a theory considered more “natural” than fine-tuning its couplings?

Theories whose behavior would qualitatively change if their couplings were not fine-tuned to particular values are often dismissed as "unnatural" (in high-energy physics) or "unrealistic" (in ...

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