All Questions
Tagged with conventions complex-numbers
48
questions
3
votes
1
answer
89
views
Equation for real/complex $\phi^4$ theory
On wikipedia (see this link), the Lagrangians of the $\phi^4$ equation for real AND complex scalar fields are given. One may derive the Klein-Gordon equation by inserting into the Euler-Lagrange-...
- 743
1
vote
1
answer
70
views
Why does Dirac bilinear $\bar{\psi}\sigma^{\mu\nu}\psi$ is frequently written with a factor of $i$?
The tensor Dirac bilinear $\bar{\psi}\sigma^{\mu\nu}\psi$ has the matrix tensor $\sigma^{\mu\nu}=\frac{i}{2}\left[\gamma^\mu,\gamma^\nu\right]$.
I can understand that the factor of $\frac{1}{2}$ is a ...
- 925
1
vote
2
answers
171
views
Why is time harmonic follow the form of $e^{-i\omega t}$, not $e^{i\omega t}$? [closed]
In physics, when we solve an PDE or ODE, the solution usually has the form of
\begin{equation}
f=C_+e^{i\lambda x}+C_-e^{-i\lambda x}
\end{equation}
and the "causility" will eliminate one ...
- 19
2
votes
0
answers
86
views
How to interpret $\int\mathrm{d}^2z$? [duplicate]
In chapter 6 of Tong's lecture notes on string theory when calculating the Virasoro-Shapiro/4-point Tachyon amplitude he arrives at the integral
\begin{align*}
C(a, b) = \int\mathrm{d}^2z\ |z|^{2a-2}|...
- 2,068
1
vote
2
answers
192
views
Quaternions as rotation generators
The following exercise appears in Geometric Algebra for Physicists by Chris Doran and Anthony Lasenby in section 1.8.
The unit quaternions $i, j, k$ are generators of rotations about their ...
- 2,599
3
votes
1
answer
59
views
Can I switch the convention of QCD by replacing coupling constant $g$ with $-g$?
There are two equivalent conventions in QCD that give two different definitions of the covariant derivative operator: ${D_\mu } = {\partial _\mu } - {\rm{i}}gA_\mu ^\alpha {T_\alpha }$ and ${D_\mu } = ...
- 41
0
votes
0
answers
61
views
Why do we prefer to use $i$ with generators in Lie Algebra [duplicate]
I am reading A. Zee. Group theory in a Nutshell for Physicists and for some reason, he prefers to write the generators with an $i$ near them
For example, a rotation can simply be described as:
$$e^{\...
- 1,324
0
votes
1
answer
103
views
Arbitrarity of $i$ in the propagator
My question is simple: how arbitrary can the factor in front of the propagator be?
What I mean by that is, if we call the wave operator $K$ and the propagator $G$, I've seen different books use ...
- 5,931
2
votes
1
answer
642
views
Complex valued Grassmann variables $(\theta \eta)^* $, $(\theta \eta)^T$ and $(\theta \eta)^\dagger$
Since hermitian conjugation and complex conjugation are serious issues in a QFT lagrangian with Grassmann variables, see here and here. Let us try to go to the bottom.
We start by accepting the ...
- 593
0
votes
1
answer
255
views
Factor $\frac{1}{2}$ in scalar kinetic Lagrangian in QFT [duplicate]
Why is it that sometimes I see kinetic term of scalar Lagrangians written like this $$\mathcal{L}=\partial_\mu\phi^\dagger\partial^\mu\phi+\dots$$ like for example in scalar electrodynamics, while ...
- 5,931
0
votes
1
answer
786
views
The Generators of $SO(3)$, with an extra "$i$" [duplicate]
I am studying group theory by myself and while i was reading "Physics from symmetry", Jakob Schwichtenberg's book, he said it was conventional in physics to write the generators of $SO(3)$ ...
- 3
0
votes
0
answers
43
views
Definition of $\mathfrak su(2)$ Lie algebra in Physics vs Mathematics [duplicate]
In B.C.Hall mathematics book on Lie Groups the Lie algebra is by definition closed under the Lie bracket operation, that is $[X, Y]\in \mathfrak g$ for every $X,Y \in \mathfrak g$.
The $\mathfrak su(...
- 613
0
votes
1
answer
312
views
Why does the Riesz Representation Theorem establish an antilinear correspondence between bras and kets?
I am reading Ballentine's Quantum Mechanics and, in the mathematical preqrequisites, he asserts that, by construction, the Riesz Representation Theorem establishes an antilinear correspondence between ...
- 1,095
1
vote
0
answers
56
views
$i$ in Green functions
This sounds like a dumb question in my mind but I can't find an answer anywhere so I'll ask it anyway. Why on the Wikipedia article on Green function is written that $$LG(x,s)=\delta(x-s)$$ but on ...
- 5,931
4
votes
1
answer
111
views
Why is there an $i$ in the definition of hadronic decay constants?
The decay of (for example) a pion can be parameterized by a decay constant $f_\pi$ defined via
$$ \langle 0 | \bar d \gamma_\mu \gamma^5 u |\pi^+(p) \rangle = i f_\pi p_\mu $$ $$ f_\pi \approx 131 \...
- 131