# 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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### Tensor product of universe Hilbert space and black hole Hilbert space equals?

I'm a newbie struggling to parse concepts important in black hole, holographic principle and related issues. My query comes from Thomas Thiemann’s Modern Canonical Quantum Gen. Relativity, which ...
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### Why does time evolution preserve the norm of a wavefunction?

I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on: Since time-evolution must preserve the norm of the wave-function, it follows that $U(t', t)$ must be a ...
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### Infinite-dimensionality of unitary representations of non-compact simple Lie Groups

I have a question about the argument given in On finite-dimensional unitary representations of non-compact Lie groups. I have been looking for a good proof for this claim for a little while now. I ...
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### Interactions terms among vector fields in Lagrangian

In the QED Lagrangian we have terms that look like $\partial_{\nu} A_{\mu}$ and $J_{\nu} A_{\mu}$. We cannot have $A_{\mu} A^{\mu}$ as its coefficient would give a mass term, and for the photon we ...
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### The necessity of ground state in QFT

Why we always want to have a ground state in every physical theory? For example, when we try to quantize Dirac Hamiltonian and encounter a Hamiltonian without a ground state, we take a step back and ...
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### Show $S$-operator is unitary

In an exercise, we are supposed to show that the scattering matrix on the right of $$S_1(E)= \begin{pmatrix}t_1 & r_1' \\ r_1& t_1'\end{pmatrix}\delta(E_f-E_i)$$ is unitary. We are explicitly ...
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### On finite-dimensional unitary representations of non-compact Lie groups

In this thread Lorentz transformations for spinors, V. Moretti made a claim as follows: "it is possible to prove that no non-trivial finite-dimensional unitary representation exists for a non-compact ...
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### Lorentz transformations for spinors

The lorentz transform for spinors is not unitary, that is $S(\Lambda)^{\dagger}\neq S(\Lambda)^{-1}$. I understand that this is because it is impossible to choose a representation of the Clifford ...
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### Understanding how the quartic Higgs coupling turns negative at high energies

How is the conclusion that the Higgs quartic coupling becomes negative at high energies ($\sim 10^{10}$) GeV inferred? I'm looking for a reference that has the necessary idea and calculations to ...
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### Why is time-evolution operator unitary?

When we shift the system's time from $t=0$ to $t = t$, we can define the following operator $\hat{U}$. $$\hat{U} = e^{- i \hat{H} t / \hbar} \, .\tag{1}$$ So many (as far as I read, almost all of) ...
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### Assumptions in the proofs for the optimality of Grover's Search Algorithm

I was trying to understand this paper in which it is proved that Grover's search algorithm is optimal. On page 4, beginning of section 2 of the paper the author says the following In the proof I ...
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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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### Can we make sense of a Hamiltonian $a^\dagger a^\dagger + a a$?

If I have a Hamiltonian given by $$H = a^\dagger a^\dagger + a a$$ where, $[a,a^\dagger] = 1$, Can I make sense of it, by generalizing the notion of vacuum? If not what sort of troubles I would run ...