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.
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Why would the normalisation of a vector with 2 complex numbers only give us 3 real parameters?
If a general spin state is characterised by 2 complex numbers, that would mean that 4 real parameters characterise it. I'm assuming that they are given by the complex and real parts of said complex ...
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Conditions for a transformation to be time reversal invariant
In Elements of Nonequilibrium Statistical Mechanics, Balakrishnan writes:
The following question then arises naturally: All these ‘irreversible’
equations must originate from more fundamental ...
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'Principles of Quantum Mechanics- 4th Edition' - Dirac / Request for help with 'simple' formula
As an amateur who read and enjoyed the integral 'Feynman Lectures on Physics', I am now reading Dirac's 'Principles'.
In §17 - The representation of linear operators - I struggled with a rather '...
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Boost generators and boost representation in $SO(3,1)$
The book "Groups, Representations and Physics" by Jones says on p.208
[...] the generators of boosts $\textbf{Y}$ are anti-Hermitian. Hence when exponentiated they produce anti-unitary ...
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Why does the imaginary unit $i$ appear in the Schrödinger equation? [duplicate]
In the time-dependent Schrödinger equation
$$i(h/2π)(dΨ/dt)=ĤΨ,$$
the imaginary unit $i$ plays a central role.
I understand that it is related to complex wavefunctions, but I would like to understand ...
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Wick rotation of the Polyakov action in Polchinski: contour deformation and convergence
On p.82 in String Theory, vol I, section 3.2, Polchinski gives a formal argument of the equivalence of the Polyakov action in Minkowskian and Euclidean form
$$S=-\frac 1{4\pi\alpha'}\int d\tau d\sigma ...
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Conserved Charge in Quantum Complex Scalar Field
I have been following an introductory quantim field theory course,* and studying the free complex scalar field theory, we find the conserved charge $Q$ has an ordering ambiguity. But I don't ...
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Why complex conjugated classical field gets replaced by adjoint?
A complex classical field, $\phi(t,\vec{x})$, is represented by an operator $\hat{\phi}(t,\vec{x})$ in QFT. Why is its complex conjugate, $\phi^*(t,\vec{x})$, gets replaced by the adjoint operator, $\...
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Importance of relative phase in quantum mechanics [duplicate]
I am a Math student learning Quantum Mechanics. I know that, in Quantum Mechanics, all states are defined up to a global phase, that is: if $|\psi\rangle$ is a state, then $e^{i\theta}|\psi\rangle$ ...
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What kind of subspace is this?
(Original title "When is a subspace called real?" was changed after discussion)
This is perhaps partly about math terminology, but the situation often occurs in physics so I'd like to know ...
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Riemann 2-manifold volume form: complex vs real
Question 1: is it always possible to write the metric in that form? Is it sufficient the local conformally-flat form to obtain the volume?
Question 2: Is the volume form in (4.1) well-defined? Going ...
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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 ...
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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 ...
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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)}{...
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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}) \...
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