# 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 isn't the Segal-Bargmann space used more often in Quantum Mechanics?

The Stone-von Neumann Theorem states that a Hilbert space on which is defined an irreducible set of operators which satisfy the exponentiated canonical commutation relations is unitarily equivalent to ...
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### In light of Wick rotations is position and time on the same footing in QFT?

Taken from here Wick rotation connects statistical mechanics to quantum mechanics by replacing inverse temperature $1/(k_{B}T)$ with imaginary time $it/ℏ$ But I was under the impression position ...
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### Is complex conjugation operator Hermitian?

I wonder whether the complex conjugation operator, defined on a wavefunction as $$C \psi(x) = \psi^*(x),$$ is Hermitian? On one hand, its eigenvalues are not necessarily real. On the other ...
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### Is there any "real" use of complex analysis in quantum mechanics? [closed]

After learning some quantum mechanics, I see a lot of applications of complex numbers. However, I have not yet seen any application of complex analysis. The full name for complex analysis is "...
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I was trying to prove the identity $\overline{\displaystyle{\not}{a}\displaystyle{\not}{b}\dots \displaystyle{\not}{p}} = \displaystyle{\not}{p}\dots \displaystyle{\not}{b}\displaystyle{\not}{a}$. On ...
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### What significance do field-operators have, if they don't correspond to observables because of non-hermicity?

Since field-operators are not always hermitian (for example in case of a complex scalar field, or the dirac-field), they don't (in the quantum-mechanical sense) correspond to observables. Does that ...
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### Proving that $$[\phi(\vec{x}, 0), \phi(\vec{x}, t)] \sim e^{-i m t}-e^{+i m t}$$ in QFT

So far, I get the following (for the left term in the integral, $d$=3): \begin{aligned} \Delta_{+}(x) &= \int \frac{\mathrm{d} \vec{p}^{d}}{(2 \pi)^{d} 2 e(\vec{p})} \exp (-i t e(...
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### Complex numbers in physics [duplicate]

Can someone please explain the origin of complex numbers in physical values. For instance, denoting a plane wave with Euler's identity and also the complex relative permittivity? Thank you.
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### Can there be an **essential topic** in physics which cannot be archimedean? [closed]

In physics it seems everything is explained with $\mathbb R$ or $\mathbb C$ typed entitites. Is there anything in or that would be in future in physics that would need the utility of $p$-adics in an ...
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### "Imaginary-time" argument in high energy physics

In many-body physics, there are many "imaginary-time" techniques, such as Matsubara Green's function, imaginary-time path integral and others. It seems that these concepts are frequently used in ...
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