2
votes
1answer
55 views

Connection between conserved charge and the generator of a symmetry

I'm trying to understand the connection between Noether charges and symmetry generators a little better. In Schwartz QFT book, chapter 28.2, he states that the Noether charge $Q$ generates the ...
1
vote
0answers
55 views

What's the value of the coupling constant in interacting field theories?

Consider this Lagrangian : $L = \frac{1}{2}(\partial_\mu \Phi)^2 - \frac{M^2}{2}\Phi^2 +\frac{1}{2}(\partial_\mu \phi)^2 -\frac{m^2}{2} \phi^2 -\mu\Phi\phi^2$ Its interaction term is given by : ...
0
votes
0answers
113 views

Derivation of (2.45) in Peskin and Schroeder

I'm having trouble understanding the step $$\left[\pi (\vec{x},t),\int d^{3}y ~(\frac{1}{2} \pi (\vec{y},t)^{2}+\frac{1}{2}\phi (\vec{y},t)(-\nabla^{2} +m^{2})\phi (\vec{y},t)) \right]$$ $$ =\int ...
2
votes
1answer
88 views

Schrödinger evolution for a Klein-Gordon equation

I have a problem with the transition from quantum relativistic wave equations (specifically Klein-Gordon equation) to QFT, since a lot of assumptions seem implicit. For example I have a problem with ...
1
vote
0answers
53 views

path integral quantization of EM field derived from canonical quantization?

In Peskin's QFT book page 294, he formally addressed the quantization of EM field, $$propagotor_{EM}=\frac{-ig_{\mu\nu}}{k^2+i\epsilon}$$ Now that we have the functional integral quantization ...
3
votes
1answer
173 views

Finding the creation/annihilation operators

Using Minkowski signature $(+,-,-,-)$, for the Lagrangian density $${\cal L}=\partial_{\mu}\phi\partial^{\mu}\phi^{\dagger}-m^2\phi \phi^{\dagger}$$ of the complex scalar field, we have the field ...
4
votes
2answers
181 views

Hamilton formalism for Dirac spinors

Let's have the Dirac free lagrangian: $$ L = \bar {\Psi} (i\gamma^{\mu}\partial_{\mu} - m) \Psi . $$ I can rewrite it as $$ L = i\Psi^{\dagger}\partial_{0}\Psi - H_{d}, \quad H_{d} = ...
10
votes
2answers
205 views

Conservation of phase space volume in Rindler space-time

Let us consider Rindler space-time, i.e. Minkowski space-time as seen by a constantly accelerating observer. My question is, does Liouville's theorem, i.e. the conservation of phase space volume in ...
2
votes
3answers
299 views

EQUAL TIME commutation relations

Why is equal time commutation relation used in canonical quantization of free fields?
3
votes
2answers
805 views

Gauge Invariance of the Hamiltonian of the electromagnetic field

The Hamiltonian for an electron of mass $m$ and charge $e$ in an exterior electromagnetic field is $$H=\frac{1}{2m}(p-(e/c)A)^2+e\varphi.$$ The corresponding (via canonical quantization) quantum ...
8
votes
1answer
282 views

The most general procedure for quantization

I recently read the following passage on page 137 in volume I of 'Quantum Fields and Strings: A course for Mathematicians' by Pierre Deligne and others (note that I am no mathematician and have not ...
2
votes
1answer
144 views

Does a Lagrangian imply a well-defined quantum Hamiltonianian with a Hilbert space?

The question is about: (1) whether giving a Lagrangian is sufficient enough to (uniquely) well-define a Hamiltonianian quantum theory with a Hilbert space? The answer should be Yes, or No. If ...
4
votes
1answer
437 views

Why does Quantum Field Theory use Lagrangians rather than Hamiltonains? [duplicate]

Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains? I heard many reasons, but I'm not sure which is true. Some say it's just a matter of beauty, so Lagrangians are more ...
5
votes
1answer
175 views

Quantization surface in QFT

What does the Quantization Surface mean here? Reference: H. Latal W. Schweiger (Eds.) - Methods of Quantization
2
votes
1answer
391 views

Deriving the Hamiltonian density for a free scalar field

I'm working through my old notes on QFT (cf. Ref 1) and I'm not quite sure how to approach the derivation of the Hamiltonian density for a free scalar field (question 2.3 on page 19) and the ...
1
vote
1answer
122 views

Formulation of the uncertainty principle for a system?

There is a biological system that I can indeed describe by a simple quantum Hamiltonian $H$ having eigenstates $|q\rangle$ labelled by the numbers $q$, and having energies proportional to $f(q)$ - ...
4
votes
1answer
174 views

Does the Renormalization of QFT Contradict Canonical Quantization?

Does the renormalization of QFT contradict canonical quantization? In canonical quantization, you take the classical fields and canonical momenta and turn them into operators, and you require that ...
1
vote
0answers
228 views

Gradient involved commutator in $\phi^4$ theory

In a phi fourth theory, the Hamiltonian density is: $$\mathcal{H}=\frac{1}{2}\pi^2+\frac{1}{2}(\nabla \phi)^2+\frac{1}{2}m^2\phi^2+\frac{\lambda}{4!}\phi^4$$ Now I impose the usual equal time ...
2
votes
2answers
196 views

Heisenberg evolution equation for $\hat{\phi}$

Consider quantum Hamiltonian of free massive scalar particle: $$\hat{H} = \int d^3x \left[\frac{1}{2} \hat{\pi}^2 (t, \vec{x}) + \frac{1}{2} \partial_i \hat{\phi}(t, \vec{x}) \partial_i \hat{\phi}(t, ...
2
votes
4answers
551 views

Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
4
votes
3answers
518 views

Dirac equation as Hamiltonian system

Let us consider Dirac equation $$(i\gamma^\mu\partial_\mu -m)\psi =0$$ as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that ...
2
votes
3answers
180 views

Quantizing first-class constraints for open algebras: can Hermiticity and noncommutativity coexist?

An open algebra for a collection of first-class constraints, $G_a$, $a=1,\cdots, r$, is given by the Poisson bracket $\{ G_a, G_b \} = {f_{ab}}^c[\phi] G_c$ classically, where the structure constants ...
3
votes
2answers
202 views

Ordering Ambiguity in Quantum Hamiltonian

While dealing with General Sigma models (See e.g. Ref. 1) $$\tag{10.67} S ~=~ \frac{1}{2}\int \! dt ~g_{ij}(X) \dot{X^i} \dot{X^j}, $$ where the Riemann metric can be expanded as, $$\tag{10.68} ...
6
votes
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 ...
2
votes
1answer
255 views

Equivalence of classical and quantized equation of motion for a free field

Suppose a classical free field $\phi$ has a dynamic given in Poisson bracket form by $\partial_o\phi=\{H, \phi\}$. If we promote this field to an operator field, the dynamic after canonical ...
3
votes
3answers
559 views

What is the difference between manifest Lorentz invariance and canonical Lorentz invariance?

I often read that the Lorentz symmetry is manifest in the path integral formulation but is not in the canonical quantization - what does this really mean?
1
vote
0answers
199 views

How important are constrained Hamiltonian dynamics and BRST transformation as a formalism?

I am to study BRST transformations, for which I'm currently trying to understand constrained Hamiltonian dynamics to treat systems with singular Lagrangians. The crude recipe followed is Lagrangian -> ...
19
votes
5answers
3k views

Why not using Lagrangian, instead of Hamiltonian, in non relativistic QM?

When we studied classical mechanics on the undergraduate level, on the level of Taylor, we covered Hamiltonian as well as Lagrangian mechanics. Now when we studied QM, on the level of Griffiths, we ...
4
votes
2answers
1k views

To what extent is the “minimal substitution” or “minimal coupling” for the EM vector potential valid?

In all text books (and papers for that matter) about QFT and the classical limit of relativistic equations, one comes across the "minimal substitution" to introduce the magnetic potential into the ...