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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The unitary operation of the Schrödinger cat example
If the initial state of the system is:
$|\psi(0)\rangle$ = $|0_{cat}0_{gas}\rangle$
Where $|00\rangle$ is the state where the gas has yet to decay and the cat is still alive. Then say the state ...
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Positive Definiteness of Killing Form in Gauge Theory
This question is related to requirement that the gauge group of a gauge theory be a direct product of compact simple groups and $U(1)$ factors but is not the same as, for example, this question (...
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Help proving bound on POVM measurement probabilities
I am trying to follow Nielsen and Chuang's 1 proof that the difference in measurement probabilities is bounded by the difference between two unitary operators applied to a given state.
Can someone ...
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How do I find the state and the energy exchange of a system after time $t$?
Assuming that the Hamiltonian of my system is of the form:
$$H=k\left( \left| 0\right> \left<1 \right| + \left| 1\right> \left< 0\right| \right)$$
and that my initial state can be ...
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1answer
61 views
Sign in infinitesimal Unitary transformations
When studying unitary transformations in QM, most of the textbooks I used defined the unitary transformation operator as
$$ \hat{U}(\alpha) = e^{-i \alpha \hat{G}} \, \, , $$
where $\alpha$ is the ...
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Determining if a process is information preserving in quantum mechanics
I was wondering if one is provided with two quantum states such that one is the evolution of another through some unknown process, is it possible to determine if the aforementioned process is ...
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135 views
Infinitesimal generator of change of basis (Fock Space)
I'm trying to find unitary transformation and prove that the infinitesimal generator for a change of basis with spatial depedency $$|\vec{r} \rangle \rightarrow e^{i \theta (\vec{r}) }|\vec{r}\rangle ...
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What does QFT say about non-linear processes?
In QFT one can use the S matrix theory. We have a IN free system in the far past. It interacts in a black box in the present and there is a free OUT system in the far future.
We have OUT = S IN with ...
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Prove that the scattering operator is unitary [duplicate]
Let $H_0$ be some initial time-independent hamiltonian, and let $V$ be a scattering potential, such that the hamiltonian of a scattering process is:
$$H=H_0+V$$
Define the quantum states $|\psi_i\...
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35 views
A unitary transformation induces a change $\delta\alpha$ on an operator that commutes with a complete set. Why must $\delta\alpha\propto 1$?
I'm reading Schwinger's 1951 paper "The Theory of Quantized Fields I". He described $\alpha$ as a complete set of commuting Hermitian operators, and considers a unitary transformation of ...
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Is entanglement a non-unitary transformation?
My question is motivated by the facts that measurement is a non-unitary transformation, and entanglement is necessary for measurement.
To clarify: I mean to say the transformation associated with the ...
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Irreducible unitary representation of $\text{SU}(2)$ and multiplicity of $J_3$ eigenvalues
I searched a lot about this, but I can't find anything near to an answer.
I'm trying to find an irreducible unitary representation of $\text{SU}(2)$ Lie group, so writing the generic element as $e^{i\...
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What's the motivation behind the seemingly artificial setup of the no-deleting theorem?
If you asked me what physical result would be naturally referred to as "the no-deleting theorem", then I would probably guess something like this:
Given a designated "blank" state ...
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Unique properties of Hamiltonian
Given a general (time-independent) system where I have some Hermitian operator $O$, is there a way of knowing if $O$ happens to be the Hamiltonian?
In other words, are there special mathematical ...
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How do unitarity and locality break in theories involing gravity?
In the science-pop article A Jewel at the Heart of Quantum Physics https://www.quantamagazine.org/physicists-discover-geometry-underlying-particle-physics-20130917/, there is a footnote, with the ...
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72 views
Most general Gauge Lie group in a Yang-Mills theory
Mathematicians have done a complete classification of all possible Lie groups. Is there a set of conditions that allows us to identify which Lie groups from the classification can possibly act as a ...
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Lorentz transformation in QFT
A Lorentz transformation can be seen as a change in reference frame. So, after apply a Lorentz transformation to a system (or change the reference frame), how should the state and field operator ...
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Is $U^\dagger(R)\hat{H}U(R)=\hat{H}$ always true?
Consider a Rotation transformation on momentum state,
$$U^\dagger(R)\hat{\mathbf{p}}U(R)=R\hat{\mathbf{p}}$$
Now the question is whether,
$$U^\dagger(R)\hat{H}U(R)=\hat{H}\,?$$
Here, $\hat{H}$ is the ...
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3answers
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Does QED evolve unitarily after the Schwinger limit?
If QED becomes nonlinear after the Schwinger limit, shouldn't QED no longer be unitary (above the limit) since linearity is a requirement of a unitary operator (and vice versa)?
Does this mean that ...
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1answer
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Unitarity and amplitudes
In Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory there are few statements about analytical structure of amplitudes.
I want to understand statement:
In a local theory of ...
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Why is the Weyl tensor square gravity non-unitary?
I want to understand constraints of unitarity in quantum field theory. There is quite folklore statement:
gravity with a Weyl tensor square term is non-unitary
How to understand this? (I think that ...
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The Information Paradox and the Copenhagen Interpretation
I'm curious why the so-called 'information paradox' is a paradox. I know that it's regarded as a paradox due to the fact that Hawking radiation seemingly violates the unitarity of the time-evolution ...
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For which values of lambda the euclidean two-point function $(p^2 +m^2)^{-\lambda}$ is reflection positive
The case $\lambda=1$ is well known free field kernel. What about $\lambda$ in between 0 and 1 ??
... for $\lambda>1$ I have a proof that the kernel is not reflection positive ,
...
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How was information generated in the early universe?
Since according to the unitarity principle of quantum mechanics the total amount of information in the universe is conserved over time, how was this information generated in the first place? And how ...
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Negative sign of kinetic term in Lagrangian in Schwartz's book
In the book Quantum Field Theory and the Standard Model by M. D. Schwartz, the author has used negative kinetic term in most of the Lagrangians. See equation (7.91) at page 97, for example. (Focus on ...
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How to find the unitary operation of a depolarizing channel? [duplicate]
Suppose we have a depolarizing channel operation
$$E(\rho)=\frac{p}{2}\textbf{1}+(1-p)\rho$$
acting on a Spin$\frac{1}{2}$ density matrix of the form $\rho=\frac{1}{2}(\textbf{1}+\textbf{s}\cdot\...
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1answer
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Alternative Physics - Attractive Electromagnetism / Vector Gravity
If you flip the sign of the term containing the field strength tensor (e.g. change $-\frac{1}{4}F^{\alpha \beta}F_{\alpha \beta}$ to its negative) in the Lagrangian for electromagnetism, you get a ...
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Finding the unitary transformation associated with a symmetry
Suppose we have a Hamiltonian that has a symmetry. Let's consider a simple example where the Hamiltonian depends on a vector $ H(\vec r)$ such that a rotation $R \vec r$ is a symmetry.
So, $H(r)$ and $...
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1answer
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Conservation of Distinctions in Quantum Mechanics
Recently I have been reading Quantum Mechanics The Theoretical Minimum by Leonard Susskind. In the book he mentions the law of conservation of distinctions, i.e. the conservation of information.
He ...
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1answer
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Unitarity implies branch cuts in $s > 4m^2$ in the $S$-matrix
Why does unitarity imply a branch cut in the $S$-matrix after $s > 4m^2$ where $s$ is the Mandlestam variable and $m$ is the mass of the particle? Assume identical particle scattering.
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Proof that $U$ is an unitary operator [closed]
I have a function $f$ mapping a bit onto another bit, i.e. $f : \{0, 1\} \rightarrow \{0, 1\} $. The function f is either constant,
so f(0) = f(1) or balanced, so f(0) $\neq$ $f$(1). The quantum gate ...
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1answer
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Are there finite-dimensional unitary irreducible representations in Euclidean space?
One can show that there are no unitary finite-dimensional irreducible representations of the Poincaré group. The reason is that the generators for boosts are antihermitian.
Since boosts are basically ...
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Analytic Continuation of the $S$-Matrix and Analytic Continuation of Spinors
Broadly speaking, what is actually being done when one analytically continues the external particles' momenta in scattering amplitudes in QFT?
Some associated more detailed questions:
When thinking of ...
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1answer
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Positive mass but negative/positive/neutral charge
I have heard that the mass of a particle can't be negative because the hamiltonian should be bounded from below. What is the formal argument regarding this and why the same argument doesn't follow ...
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Can be a Schmidt Decomposition Written in this Form?
The Schmidt Decomposition of a state $|\Psi_{AB}\rangle = \sum C_{ij}|i_A\rangle |j_B\rangle$ can be written in this form $|\Psi_{AB}\rangle = \sum \lambda_k|k_A\rangle |k_B\rangle$ where $\lambda_k$ ...
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Weinberg derivation of Lie Algebra [duplicate]
In the derivation of the Lie algebra in the first volume of Quantum Theory of Fileds by Weinberg, it is assumed that the operator $U(T(\theta)))$ in equation (2.2.17) is unitary, and the rhs of the ...
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3answers
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Isometric equivalence of purifications of quantum states
Following the notes here (Quantum Information Theory Tips 5 at ETH), we state the following result. For any quantum state $\rho_A$ and purifications $\vert\psi\rangle_{AB}$ and $\vert\phi\rangle_{AC}$,...
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2answers
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Connection between isometries and projectors in QM
I realize this question is technically a mathematical one but I think it is seen often enough in quantum information so I ask it here. The following is the definition of an isometry in Mark Wilde's ...
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1answer
90 views
Under what assumptions does a state following the TDSE converge to its ground state?
Until $t=0$ a system is in an eigenstate $\psi_0(x)$ of the Hamiltonian $\hat{H}_0$. The time-evolution is the trivial phase factor. Now at $t=0$ the system changes to $\hat{H}$ (you can assume it is ...
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2answers
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Physical interpretation of Lorentz group non-compactness in the case of Weyl spinors
If we consider the generators of Lorentz group $J$ and $K$, it is possible to indroduce the operators $J^{\pm}=\frac{J\pm iK}{2}$ which shows the $SU(2)\times SU(2)$ structure of the Lorentz group. ...
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Must all operators be Unitary? [closed]
Do all operators on a quantum state have to be unitary? Does this mean that there are non-unitary operators that act on quantum states in real life?
Context:
Let's say we have a composite quantum ...
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1answer
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Is it possible to build a non-unitary quantum circuit?
Is it possible to build a non-unitary circuit of quantum Boolean gates, apart from using measurement to achieve this?
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1answer
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Deriving Unitarity of $S$-matrix in 1D Quantum Mechanics
So, I was studying about scattering across a one-dimensional unknown potential ( pretty elementary Quantum Mechanics) and how if we know the $S$-matrix of such a system, we can deduce an awful lot of ...
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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 $S$-...
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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, ...
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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 ...
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1answer
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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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1answer
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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 $a$. ...
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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 ...
