# 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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### Uniform dynamics in quantum mechanics

I've found in the book "Quantum Processes System & Information" of Benjamin Schumacher the following definition of "uniform dynamic": it often happens that the basic dynamical ...
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### Relationship between unitaries generated by a Hamiltonian and its negative sign

Consider two unitary operations $U_1$ and $U_2$ defined by: $\partial_t U_1 = -iH_1U_1$ and $\partial_t U_2 = iH_1U_2$ Here, $U_1$ is generated by $H_1$ and $U_2$ is generated by $-H_1$, with the ...
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### Fermi theory cutoff from unitarity bound

Tree-level cross sections for processes described by Fermi theory behave like $\sigma$ $\sim$ $G_{F}^2 \cdot s$, where $G_{F}$ is the Fermi constant and $\sqrt s$ is the energy entering in the ...
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### Motivation behind reflection positivity

I have taken a look at this physicsSE question, Wikipedia, and this paper by Jaffe which go over reflection positivity. While they all nicely explain the definition behind reflection positivity and ...
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### Translation operators and positive-semidefinite condition

Good day. I have an operator $\exp{(-i\mu\hat{q})}\exp{(-i\nu\hat{p})}$, where $\hat{q}$, $\hat{p}$ are the position and momentum operators, respectively. The parameters $\mu,\nu$ are some real ...
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### Inverting an integral relation involving a set of continuous modes

Consider a continuous system of bosonic modes $a(x)$, and the transformation $$U a^\dagger(x) U^\dagger = \int A(x,y)a^\dagger(y) \ dy.$$ My goal is to find $U^\dagger a^\dagger(x) U$ in terms of a ...
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### Does the Hamiltonian always commute with the Time Evolution Operator?

The time evolution operator $U(t, t_0)$ is given as the solution of the equation $$i\hbar \dfrac{\text{d}}{\text{d}t} U(t, t_0) = HU(t, t_0)$$ whether or not the system is conservative. When the ...
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### Is stochasticity totally incompatible with current theories of conservation of information in quantum physics?

Most physicists accept the unitarity principle of the universe, according to which the state of a system at any given time must determine its state at any other time (information is never lost, ...
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### How does path integral quantization ensure unitarity?

Unitarity can be verified post hoc by examining the optical theorem. In the context of path integral quantization where formal derivation starting from canonical quantization is unavailable, is it ...
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### Unitary Representation of $\text{SO}(3)$ in Position Representation

Let $R\in\text{SO}(3)$ be an arbitrary rotation, and let $U_R$ be the unitary representation of $R$ on some Hilbert space $\mathcal H$. To me, the defining property of $U_R$ is how it conjugates the ...
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In Quantum Mechanics, for coherent states $|z\rangle$ it can be prooved that if $|0\rangle$ is the vacuum state for an harmonic oscillator, therefore: |z\rangle=e^{za^{\dagger}-z^*a}|...