# 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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### Normalization of the harmonic oscillator propagator

The propagator of a quantum system is defined by $$\mathcal{K}(t,x;\,t_{0},x_{0})\,\equiv\,\left\langle x\right|\hat{U}(t,\,t_{0})\left|x_{0}\right\rangle.$$ In this notation, the unitarity demands ...
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### Why do we expect unitarity to be preserved in the black hole information paradox?

Consider the following way of describing the black hole information paradox: Suppose we start with a pure quantum state and a black hole of mass $M$. Now we throw the pure state into the black hole ...
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### Can you apply non-unitary operators to a qubit?

I am wondering if it is possible to apply continuous, invertible transformations to a qubit which are not linear, i.e. not elements of $U(N)$ where $N=2^n$ where we have $n$ qubits. Consider $n=1$. ...
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### Measurement problem and precise mathematical calculation

The infamous measurement problem is a problem in the foundations of quantum mechanics: different people have different views how to understand this problem: some people even deny that there is any ...
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### Why must the propagator exponent be imaginary?

In response to asmaier's question, qmechanic showed why the propagator must be $\exp(cS)$. That made perfect sense. But can it also be shown that $c$ is imaginary? I believe it follows from ...
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### Basis change in the context of the Schroedinger and Heisenberg picture

I am trying to understand the following statement: One sometimes summarizes this situation with the slogan: In the Schr¨odinger picture the basis vectors (provided by the observables) are fixed, while ...
1 vote
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### How is the Wigner little group representation of Poincaré group Unitary?

From Weinberg's QFT Vol.1, eq(2.5.11): $$U(\Lambda)\Psi_{p,\sigma}=({N(p)\over N(\Lambda p)})\sum_{\sigma'}D_{\sigma'\sigma}(W(\Lambda,p))\Psi_{\Lambda p ,\sigma '}.\tag{2.5.11}$$ However, this is not ...
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### Proof that different representations of gamma matrices are connected by a unitary transformation

The different representations of gamma matrices are related by a similarity transformation \begin{equation} \gamma^{\mu'}=S\gamma^{\mu}S^{-1} \end{equation} for some non-singular $S$. I have to prove ...
1 vote
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### Is there a physical reason why the time evolution in quantum mechanics is given by $e^{-itH}$? [duplicate]

Let $H$ be the Hamiltonian operator. Since $H$ is self-adjoint, by Stone's theorem there is a strongly continuous one-parameter unitary group $U(t)$ such that $U(t) = e^{-itH}$. Mathematically this ...
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### Is unitary time evolution the same as obeying the Schrödinger equation?

In this question, the answer says that unitary time evolution means that probability is conserved. Is this the same as saying that a system obeys the Schrödinger equation?
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### Is linearity sufficient to guarantee probabilities are invariant under change of reference frame?

In his Symmetry Principles in Quantum Physics, Fonda seems to write (in a footnote at the bottom of page 13) that if a bijection $T$ on a vector space is linear or antilinear, then for any $\phi,\psi$ ...
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### Integral of the square of the spectral density in a quantum field theory

The quantity of interest is $$\int_0^\infty dE \, \rho(E)^2$$ where $\rho(E)$ is the spectral density in a Quantum Field Theory. I am wondering whether it has any physical meaning, and it can be ...
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### Is unitarity equivalent to linearity plus conservation of the norm?

Unitarity is the condition that the inner product in the Hilbert space is preserved. But if you suppose that the norm of any state is already preserved, then does unitarity follow from linearity? ...
1 vote
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### Are the generators of the $SU(2)_L\times SU(2)_R$ group unitary?

In the chiral model, the quark field $q=\begin{pmatrix} u \\ d \end {pmatrix}$ transforms like $q\rightarrow\exp({i(\theta_a\tau^a+\gamma_5\beta_a\tau^a)})\;q$. Now, I understand that the ...
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### The use of unitary operators in translating between reference frames in quantum mechanics

In Chapter 2.6 of his Lectures on groups and vector spaces for physicists, Isham describes how different observers (different reference frames) must describe the same quantum system. Consider two ...
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### Can a non-unitary process be made unitary using a transformation or by expanding the phase space?

Suppose I have a matrix differential equation: $$\frac{d\mathbf{x}}{dt} = A\mathbf{x}$$ The solution to this is $$\mathbf{x}(t) = e^{At}\mathbf{x}(0)$$ If $A^{\dagger}=-A$, then the time evolution ...
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### Uniform losses commute with linear optics: how does it work?

In many papers about quantum optics and interferometry, it's assumed or said that "it's well known" that linear optics commutes with uniform losses. In particular if we have a beam splitter ...
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### Dimensionality of unitary representation of Lorentz group

I am reading the textbook "Lectures on Quantum Field Theory", second edition by Ashok Das. On page 137, he talks about the non-unitarity of the finite-dimensional representation of the ...
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### States created by local unitaries in QFT

In quantum field theory, consider acting on the vacuum with a local unitary operator that belongs to the local operator algebra associated with a region. In such a way, can we obtain a state that is ...
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### Unitary representations of a Lorentz transformation

In QFT we have an action of the restricted Lorentz group which is implemented via a unitary transformation. In other words, if $\Lambda\in SO(1,3)^\uparrow$, then the corresponding unitary operator is ...
1 vote
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### Derivations about Time-Dependent Perturbation Theory (Griffiths)

I've been reading the Time-Dependent Perturbation Theory in Griffiths' book. There are many places that puzzle me a lot. The system is two-level($\psi_a,\psi_b,E_a<E_b$). Suppose the particle ...
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### Is quantum gravity compatible with unitary evolution?

I am thinking that they aren't strictly compatible. I have the following logical argument for this: The unitary evolution postulate says that the state of a system is given by a time-depending state ...
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### Is unitarity intimately "connected" to symmetry?

I have a couple of questions about unitarity and symmetries: Is unitarity connected to all fundamental symmetries? Is it linked to symmetries like CPT, Lorentz, Poincaré, diffeomorphism, ...
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### How to transform a Lindblad operator basis?

I'm trying to understand how to perform a unitary transformation on a set of traceless orthonormal Lindblad operators, following chapter 3.2.2 of The Theory of Open Quantum Systems by Breuer and ...
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### Is quantum cloning $|\psi\rangle|\psi_1\rangle|\psi_2\rangle|C\rangle\to e^{i\alpha}|\phi\rangle|\psi\rangle|\psi\rangle|C'\rangle$ prohibited?

I think the no-cloning theorem is too restrictive, as in, $$|\psi\rangle |\phi\rangle\to e^{i\alpha}|\psi\rangle|\psi\rangle \tag{1}$$ does not allow for any arbitrariness in the final state. Instead, ...
When we quantize $SU(N)$ gauge theories using the path integral formalism, we must introduce Faddeev-Popov ghosts and will appears as scalar fermions coupled to our gauge bosons in the Lagrangian of ...