1
vote
1answer
57 views

Operator Product Expansion in Massless 2D QED

In Peskin & Schroeder chapter 19 page 656, where the axial current anomaly of massless 2D QED is discussed, the authors go from: $$ ...
6
votes
1answer
170 views

Free Particle Path Integral Matsubara Frequency

I am trying to calculate $$Z = \int\limits_{\phi(\beta) = \phi(0) =0} D \phi\ e^{-\frac{1}{2} \int_0^{\beta} d\tau \dot{\phi}^2}$$ without transforming it to the Matsubara frequency space, I can ...
0
votes
0answers
89 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 ...
4
votes
1answer
75 views

Pressure and Density Using a General Lagrangian

Given a lagrangian of a form: \begin{equation}\mathcal{L}=f(\phi,\partial_{\mu}\phi\partial^{\mu}\phi)\end{equation} where $f$ is a function, I need to derive pressure and density in a FLRW universe ...
1
vote
1answer
53 views

Symmetry factor of $n$-point one-loop diagram [duplicate]

If we have a one-loop diagram in $\phi ^ 3$ scalar field theory with $n$ external lines, then what is its symmetry factor? I have drawn the diagram I am looking for, but instead of $6$ external ...
1
vote
0answers
40 views

CPT symmetries for a free Klein-Gordon equation and in minimal coupling

I'm studying for an exam on relativistic quantum mechanics and one of the issues to prepare is about symmetries of Klein-Gordon equation concerning $C$, $P$, $T$ transformations for a free field, and ...
6
votes
1answer
119 views

The problem in Sredniki's textbook: How do I calculate loop corrections for $\phi\phi\to\phi\phi$ with this Lagrangian?

The problem in Sredniki's textbook 10.5 : For a free scalar field $\psi$, the Lagrangian is $$\cal{L}= -\frac{1}{2}\partial^\mu\psi\partial_\mu\psi-\frac{1}{2}m^2\psi^2$$ Here we use the metric ...
6
votes
1answer
84 views

A question about a complex integration in Peskin's QFT textbook

In page 27 (2.52), the integration is $$\int_{-\infty}^{\infty}dp \frac{p e^{ipr}}{\sqrt{p^2+m^2}}$$ He says that there are two branch cuts starting from $\pm im$ But I learn in complex analysis ...
2
votes
1answer
105 views

Equations of motion for the Yang-Mills $SU(2)$ theory

I have an exercise for Yang-Mills theory. I can't find answer anywhere. Derive equations of motion for the Yang-Mills theory with the gauge group $SU(2)$ interacting with $SU(2)$ doublet of scalar ...
1
vote
1answer
50 views

Basic Field Calculations

I'm trying to derive the relation: $\phi(x)\phi(y)=:\phi(x)\phi(y):+\langle 0|\phi(x)\phi(y)|0 \rangle$ but struggling to see the first few steps I need to make. I've made the substitutions ...
1
vote
0answers
32 views

Is the reduction map completely positive? [duplicate]

I am struggling with proving the complete positivity of a general map ( granted it is CP ). The reduction map is defined as $$ \rho \rightarrow \mathrm{Tr}(\rho)I - \rho $$ It is a trivial job to ...
2
votes
2answers
161 views

Angular Momentum Operator in Quantum Field Theory

I'm following along with David Tong's QFT course and am trying to derive the result shown in question 6 on his 2nd problem sheet, but am running into problems when applying it to the free real scalar ...
1
vote
0answers
83 views

How to calculate these integrals about propagator of QFT analytically?

How to get these three analytical solutions? Thanks very much! $$ G_{ret}(x,y) = \lim_{\epsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{(p_0+i\epsilon)^2 - \vec{p}^2 - m^2} = ...
3
votes
1answer
220 views

Three integrals in Peskin's Textbook

Peskin's QFT textbook 1.page 14 $$\int_0 ^\infty \mathrm{d}p\ p \sin px \ e^{-it\sqrt{p^2 +m^2}}$$ when $x^2\gg t^2$, how do I apply the method of stationary phase to get the book's answer. ...
1
vote
1answer
64 views

Deriving commutation relations in second quantisation

I am trying to start from: \begin{align*} [\phi(x),\pi(x')] = i\hbar\delta(x-x') \\ [\phi(x),\phi(x')] = [\pi(x),\pi(x')]=0 \end{align*} to derive: \begin{align*} [a(k),a(k')^\dagger]=\delta_{kk'}\\ ...
2
votes
1answer
135 views

Relation between Dirac spinor and its adjoint

I'm trying unsuccessfully to solve the following problem in Thomson's Modern Particle Physics: "Starting from $(\gamma^{\mu} p_{\mu} - m) u =0, $ show that the corresponding equation for the ...
3
votes
0answers
68 views

Vertex for quartic interaction of complex scalar multiplet

Since I'm new to QFT and I tend to do a lot of errors during calculations, I would like you to tell me if I got the four-point vertex of the quartic interaction with a multiplet of complex scalar ...
6
votes
1answer
213 views

Integral in $n$−dimensional euclidean space

I've asked this question in Mathematics Stack Exchange, but unfortunately there is no answer yet. I repost it because this integral comes from QFT and maybe someone here did it before or could help ...
3
votes
1answer
111 views

Peskin equation 6.38

In Peskin and Schroeder's QFT book, page 189, equation 6.38, how do they get from the first line to the second line? In particular, I am stuck on the transition from what I perceive to be: $$ ...
3
votes
1answer
123 views

Contour for Klein-Gordon field transition amplitude

In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk \frac{ke^{ik\Delta ...
0
votes
3answers
81 views

Exchange Particle between an Electron Neutrino and Neutron?

How can a neutrino turn a neutron into a proton? This is the equation, $$ \nu_e + n \to p + e^- \,.$$ If you draw the Feynnmann Diagram which I attempted here, "Diagram" there isn't an exchange that ...
0
votes
1answer
61 views

How can W+ boson turn an electron to a electron neutrino?

If you look at the Feynmann Diagram of an electron capture: Whe W+ boson turns the electron into a neutrino. How is this possible? I thought the the boson carries the positive charge and converts ...
1
vote
1answer
191 views
0
votes
1answer
81 views

Two particles state of a 1D massive scalar field

Perfectly localized states are not normalized so do not belong to the Fock space (they belong to the rigged version). Suppose we approximate localized states with gaussians, what is the mathematical ...
1
vote
0answers
75 views

$E$ and $B$ fields in Axial Gauge

I am trying to compute the $\vec{E}$ and $\vec{B}$ fields in the Axial gauge ($n \cdot \vec{A}=0$) where $n^2=1$, but I'm having trouble seeing the usefulness/how it simplifies the equations.
1
vote
0answers
128 views

Step in derivation of solution to Dirac equation for hydrogen

My text, when solving hydrogen in the Dirac equation, makes the claim $\varphi_{j m_j}^{(+)} = \frac{\mathbf{\sigma} \cdot \mathbf{x}}{r} \varphi_{j m_j}^{(-)}$ where $\varphi_{j m_j}^{(\pm)}$ are ...
8
votes
1answer
268 views

Causality for the Dirac Field

In Peskin & Schroeder page 54, they are trying to show how far they can take the idea of a commutator for the Dirac field instead of anti-commutator. To this end they are examining causality, ...
4
votes
1answer
169 views

Derivation of the quadratic form of the Dirac equation

I am asked to derive the quadratic form of the Dirac equation in an electromagnetic field, $\left[\left(i\hbar \partial - \frac{e}{c}A\right)^2 - \frac{\hbar e}{2c} \sigma^{\mu\nu} F_{\mu\nu} - ...
5
votes
0answers
190 views

The commutator of scalar field [closed]

I have a real scalar field which is given by the propagator as: $$[\phi(x),\phi(y) ] =\int \frac{d^3 p}{(2\pi)^3} \frac{1}{2E_0} (\exp(-ip\cdot (x-y)) -\exp (ip\cdot (x-y)))$$ And I am asked to show ...
2
votes
1answer
46 views

A question about verifying the transverse of electric field

I came accross a question about verifying the transverse of electric field in Peskin and Schroeder's QFT p179. Given $$ \mathcal{A}^{\mu}(\mathbf{k}) = \frac{ -e}{| \mathbf{k} | } \left( ...
1
vote
0answers
58 views

Some strange transformation [closed]

In a lecture (look at the chapter "The fermion determinant in a constant field", p. 5) I found some strange transformation, which is given by eq. 18. How to prove it? Exactly, I don't understand the ...
2
votes
1answer
138 views

Derive non-linear $\sigma$ model from a theory of SU(2) matirx

It's said in Chapter VI.4 of A. Zee's book Quantum Field Theory in a Nutshell, a theory defined as $L(U(x))=\frac{f^2}{4}Tr(\partial_{\mu}U^{\dagger}\cdot\partial^{\mu}U)$, can be write in the form of ...
3
votes
2answers
137 views

Srednicki's book on QFT

I am reading Srednicki's book on QFT and there's a thing I don't quite see in chapter 6 (Path integrals in QM) equation (6.7) is ...
1
vote
1answer
277 views

Dirac field and stress-energy tensor density

I read somewhere that stress-energy tensor density is a symmetric tensor. But if I take the Dirac Field tensor: $$T^{\mu \nu}=i \psi^\dagger \gamma^0 \gamma^\mu \partial^\nu \psi $$ How could I ...
6
votes
1answer
563 views

Lorentz Invariant Integration Measure [closed]

When we canonically quantize the scalar field in QFT, we use a Lorentz invariant integration measure given by $$\widetilde{dk} \equiv \frac{d^3k}{(2\pi)^3 2\omega(\textbf{k})}.$$ How can I show that ...
3
votes
2answers
176 views

Dimensional regularization - integral

How can I derive the following formula? $$\int d^{d+1} k \frac{e^{i K X}}{K^2} = \frac{\Gamma (d-1)}{(4\pi)^{d/2} \Gamma (d/2) |X|^{d-1}}, \quad K^2 = k_0^2 + \vec k^2, KX = k_0 \tau + \vec k \vec ...
4
votes
1answer
157 views

Conserved topological charge for d=3 Yang-Mills. G=U(2)

Consider a pure Yang-Mills lagrangian density $$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}_aF^a_{\mu\nu}$$ with gauge group $U(2)$. Take the generators for $U(2)$ to be $t_0$, $t_i \ i=1,...,3$ with ...
3
votes
0answers
134 views

Fock Subspaces and Weight Vectors

This is my first time taking a physics course (I'm a mathematics major), so I'm encountering a lot of new things, which I'm kind of expected to know. In particular, how to work with Bosons. I've got ...
2
votes
2answers
123 views

Scalar QFT Fock Space

I want to demostrate the following relation of the normal ordered product: $\Omega\equiv:\exp{\left(-\int d^3k~a^{\dagger}(k)a(k)\right)}:=|0\rangle\langle0|.$ I proved the commutation relation ...
0
votes
1answer
138 views

taking the trace

could anyone show me the first couple of steps in taking the trace of something like this, im not sure how to start. 'Tr[$\gamma($$\gamma k + $$\gamma p + $$\gamma q + m) $$\gamma ($$\gamma k + ...
1
vote
0answers
67 views

An Equality in The Calculus of Many Instantons

I am reading the review on instantons. When I tried to derive formula (2.27) on page 17, I always get the different coefficient of $gF_{mn}$ term. My calculation is just directly expanding the first ...
2
votes
1answer
344 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
0answers
46 views

Proton or electron charge in the Weinberg-Salam model?

I read Quantum Field Theory, Ryder, second edition. Relation (8.86) brings us the famous result: $e = g \sin \theta_W$ Here Ryder says tht $e$ is the proton charge. However, according to what I ...
3
votes
2answers
169 views

Space translation of operators, states, and particle densities

In Sidney Coleman's Lectures he talked about space translations such that $$\tag{1} e^{ia P}\rho(x) e^{-ia P} ~=~ \rho(x-a),$$ but when I expanded the exponentials and used the commutation relation ...
3
votes
2answers
286 views

Unitary spacetime translation operator

Srednicki writes: We can make this a little fancier by defining the unitary spacetime translation operator $$ T(a) \equiv \exp(-iP^\mu a_\mu/ \hbar) $$ Then we have $$ T(a)^{-1} \phi(x) T(a) = ...
3
votes
1answer
91 views

Interaction of an electromagnetic wave with a two level system in the domain of quantum field theory

Suppose I shine an electromagnetic wave on a two-level system. I need to describe how the system evolves in context of quantum field theory i.e. using a quantized EM field in the problem. The first ...
4
votes
2answers
354 views

How to directly calculate the infinitesimal generator of SU(2)

We commonly investigate the properties of SU(2) on the basis of SO(3). However, I want to directly calculte the infinitesimal generator of SU(2) according to the definition $$X_{i}=\frac{\partial ...
1
vote
0answers
201 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 ...
1
vote
1answer
1k views

Derivation of Dirac equation using the Lagrangian density for Dirac field

How can I derive the Dirac equation from the Lagrangian density for the Dirac field?
0
votes
0answers
75 views

Question regarding operators and cylindrical coordinates

I have the following problem in my hand: I need to arrive from the Cartesian expression $$x_{j}{\partial_{k}}x_{j}{\partial_{k}}-x_{j}{\partial_{k}}x_{k}{\partial_{j}}$$ to this expression: ...