# Questions tagged [unitarity]

In quantum mechanics, a unitary operator satisfies U<sup>†</sup> U = UU<sup>†</sup> = I, where † denotes Hermitean conjugation; such operators then specify Hilbert space automorphisms and preserve state norms, so then probability amplitudes and hence probabilities. Use for conservation of probability questions under unitary state transformations.

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### Matrix for $\pi/2$ pulse?

If we have a two-state system \[ \newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \newcommand{\mean}[1]{\...
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### Do all unitary-preserving regulators necessarily turn real loop integrals into pure imaginary numbers?

The optical theorem, which results from the unitarity of the $S$-matrix, relates the imaginary part of the forward scattering amplitude to the total cross section. When using this theorem in practice, ...
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### Perturbative proof of unitarity of $S$-matrix in QED

In any standard textbook on QFT I know it is claimed that the $S$-matrix in QED is a unitary operator. I have never seen any proof of it. This should be compared with the analogous property of $S$-...
717 views

### Why is $SU(3)$ chosen as the gauge group in QCD?

Why is $SU(3)$ chosen as the gauge group. Why not $U(3)$? Why does it even have to be unitary?
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### Stone's theorem on one-parameter unitary groups and new self-adjoint operators

I have been following the proof of the Stone's theorem on one-parameter unitary groups. The question is if the current list of self-adjoint operators used in quantum mechanics, including position, ...
124 views

### What are the theoretical / mathematical problems in discarding negative solutions of Dirac equation?

I read some Q&A about it, but my question is why Dirac was so sure that he could not discard negative energy solutions. It seems so natural that energy must be positive, that I suppose that if ...
65 views

### Proof of normalizing the wave function

So suppose we have the wave function $\phi(x,t)$ in the context of quantum mechanics, that satisfies the Schrodinger's equation. We want to see that if we normalize this function at $t=0$ then it will ...
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### Anti-commutator for annihilation and creation operators: ordering of indices

I'm trying to prove that $\{\tilde a_i,\tilde a_j^{\dagger} \}=\delta_{ij}$, by defining $\tilde a_i=\sum_j \bar U_{ji}a_j$. U is an unitary matrix and $a_i$ refers to an element of the operator $a$. ...
228 views

### Do black holes comply with the principle of unitary evolution?

Claus Kiefer, "Quantum gravity", 3rd ed., page 220/221, says in chapter 7 "Quantization of black holes": "A theory of quantum gravity should give a definite answer to the question of whether ...
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### Prove that there exists a $d \times d$ unitary matrix $U$ which cannot be decomposed as a product of fewer than $d-1$ two-level unitary matrices

This is exercise 4.38 from Nielsen and Chuang: Prove that there exists a $d \times d$ unitary matrix $U$ which cannot be decomposed as a product of fewer than $d-1$ two-level unitary matrices. If ...
Consider pure a $U(1)$ Yang-Mills action $$S=-\frac{1}{4g^2}\int d^4xF^{\mu\nu}F_{\mu\nu}$$ Usually, one calculates the corresponding partition function $\mathcal{Z}(J^{\mu})$ by adding some external ...