Questions tagged [complex-numbers]
Numbers of the form $\{z= x+ i\,y:\;x,\, y\in\mathbb{R}\}$ where $i^2 = -1$. Useful especially as quantum mechanics, where system states take complex vector values.
1,093 questions
1 vote
1 answer
113 views
Interpreting one-dimensional Newtonian mechanics using complex numbers? (via Hamiltonian mechanics)
Let the configuration space of a single "point particle" be the one-dimensional affine space $\mathbb{A}^1 \cong \mathbb{R}$, with a chosen linear coordinate chart identifying some ...
- 576
3 votes
1 answer
187 views
Formal definition of $d$-dimensional integral for complex dimension $d\in\mathbb{C}$
This question concerns the definition of dimensional regularization in quantum field theory, specifically as presented in this Wilson paper (see free version here). This operation must fulfill three ...
- 808
1 vote
1 answer
108 views
Parallel transport in Yang-Mills
I'm currently reading through David Tong's "Gauge Theory" lecture notes, and came across the following parallel transport equation: \begin{equation} i \frac{dw}{d\tau} = \frac{dx^\mu(\tau)}{...
- 150
4 votes
1 answer
122 views
Contour Integral in Friedel oscillation calculation
In Section 14 of Fetter&Walecka's Quantum Theory of Many-Particle Systems, the authors evaluate the induced charge density due to a static charge impurity: $$ \delta \langle \rho(\mathbf{x}) \...
- 747
4 votes
2 answers
192 views
Integrals in Euclidean plane with complexified coordinates?
Calculations are carried out in Euclidean plane with complexified coordinates $z,\bar{z}$ as we do in CFT. I need to derive the following: $$\int{\frac{d^2 z_1}{(z-z_1)(\bar{z_1}-\bar{w})}}=\pi\ln{|z-...
- 523
4 votes
0 answers
201 views
In the space-like propagator $D(x-y)$, does the choice of branch point in the integral change its result? [duplicate]
I am calculating the space-like propagator $D(x-y)$ and I am finally getting the integral: $$\int\limits_{-\infty}^\infty dp \frac{p}{\sqrt{p^2+m^2}} e^{ipr}.$$ Because of the square root, the branch ...
2 votes
1 answer
121 views
Current density [closed]
I have a basic question: where, in this equation has disappeared $\frac{1}{2}$: $$\mathbf{j}=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*)=\frac{\hbar}{m}Im\{\psi^*\nabla\psi\}.$$ I.e. how do I ...
- 29
1 vote
2 answers
299 views
Convergence of $E$ for a system of free fermions using Green's functions
I am following "Quantum Theory of Many-Particle Systems" by Fetter and Walecka. The expression for the total ground-state energy of a homogeneous system of fermions in a box of volume $V$ (...
- 157
5 votes
1 answer
323 views
What kind of procedure is the Wick rotation to Euclidean formulation?
I'm learning QFT via the path integral formalism. I've been struggling understanding the Wick rotation to Euclidean formulation, towards which I feel very uncomfortable. In particular I cannot find a ...
- 864
0 votes
0 answers
136 views
Complex conjugate of a fermion creation operator
I am analyzing page 70 of Peskin & Schroeder concerning the complex conjugate, where in Eq. (3.145) they write the following: \begin{equation} \begin{split} -i\gamma^2 \int \frac{d^3p}{(2\pi)^3} \...
- 769
3 votes
0 answers
122 views
Another question on Wick Rotation and field expansion
I’m sorry to be yet another person confused about Wick rotation, but it’s been all day and I’m still not sure I’ve got it right. Let’s start by considering the field expansion of a scalar field: $$ \...
- 381
-2 votes
1 answer
136 views
Equating different forms of the Schrodinger Equation from Susskind's Theoretical Minimum [closed]
I have been working through Leonard Susskind's Theoretical Minimum books as introductions to higher level physics, and in Quantum Mechanics Chapter 4 the generalised (time-dependant) Schrödinger ...
- 13
1 vote
1 answer
138 views
Analytically continuing a Matsubara sum before or after algebraic manipulations [closed]
I am computing the Matsubara sum $$S(i\omega_n) = \frac{1}{\beta}\sum_m \frac{1}{(i\omega_n - i\Omega_m +a)(\Omega_m^2 + b^2)}$$ where $\Omega_m$ is a bosonic Matsubara frequency. And, $\omega_n$ is a ...
- 4,113
0 votes
0 answers
179 views
Has quaternion imaginaries $i,j,k$ ever been used to model physical spatial dimensions?
Beyond Hamilton’s original work, have there been serious attempts to treat $i,j,k$ as physical spatial dimensions? Hamilton said he sees the quaternions as mapping space in 3D and time as 1D.
- 33
0 votes
0 answers
112 views
Suggestion of notes/books on mathematical methods regarding loops (QFT)
I am looking at loop integrals in QFT. I am getting used to the ‘usual’ tricks for loops, such as Feynman parametrization + Wick rotation. I feel less confident about integration of real radiation, ...