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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Why quantum map must be hermitian?
Quantum maps transform a density matrix into another one, Assume we are in the Hilbert space :$ H_A $
the quantum map on the density matrix $\rho_A$ living in $H_A$ is : $\mathcal{L}_A$
Why $\mathcal{...
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1answer
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How do we find a time-independent Hamiltonian that generates a given unitary transformation?
I know that for time independent Hamiltonians we can make the statement
$$U = e^{-iHt}\tag{1}$$
where $H$ is a time-independent Hamiltonian (divided by $\hbar$) and $U$ the unitary, also known as ...
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Entanglement invariant under local basis change?
Today in the lecture the professor said that if we have an entangled state between two systems A and B
$$\mid \psi_{AB} \rangle = \frac{1}{\sqrt2}(\mid 00\rangle+\mid 11\rangle)$$
There is no local ...
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0answers
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Property of time evolution operators
I am trying to show that:
$\langle0|\Pi^n_{i=1}\psi(t+t_i,x_i)|0\rangle$ is independent of time.
What I did is to write everything in terms of the Schrodinger picture operators which I know are time-...
2
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1answer
65 views
Does the wavefunction probabilities have to sum to 1? [duplicate]
In quantum mechanics we are often told that $\int |\psi(x,t)|^2 dx^3 =1$.
i.e. the probabilities have to sum to 1. And that this implies the time evolution operator is unitary.
But can't we define ...
44
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3answers
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Why is the information paradox restricted to black holes?
I am reading Hawking's "Brief answers". He complained that black holes destroy information (and was trying to find a way to avoid this). What I don't understand: Isn't deleting information quite a ...
1
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1answer
105 views
Why first-order Born Approximation doesn't satisfy optical theorem?
First-order Born Approximation in Quantum Mechanics states that scattering amplitude is a Fourier transform of potential:
$$
f(\theta) = \int d^3 r^{\prime} e^{-i (\bf k - k_i)r^{\prime}} V(r^{\prime}...
3
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1answer
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Wigner proof of the non-existence of finite unitary representation of the Lorentz group
I am reading Wigner's paper ”On unitary representations of the inhomogenous Lorentz group” (Annals of Mathematics, Vol. 40, No.1, p. 149) found here: https://www.maths.ed.ac.uk/~jmf/Teaching/Projects/...
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2answers
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What's the problem with Black Hole evaporation?
Black hole evaporation is not unitary because it takes a pure state to a mixed state. On the other hand, ordinary decay processes in Quantum Mechanics do not seem very unitary either. (For example, if ...
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3answers
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Domains of $H$ and $U(t) = \exp(-iH t )$
I am not so familiar with functional analysis. But in my impression, the Hamiltonian $H$ is often not defined everywhere on the Hilbert space. On the other hand, the time evolution operator $U(t)$, ...
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Do longitudinal and scalar have anything to do with Faddeev-Popov ghosts?
In his this book, Hatfield calls ghosts the negative states appearing in the covariant (Gupta-Bleuler) quantization prescription of the electromagnetic field (page 89). When discussing Yang-Mills ...
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1answer
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Doubt in Weinberg's book on Quantum Field Theory
In page number 59 of his book on QFT, Weinberg mentions that for the operator $U$, defined for infinitesimal parameters $\omega$ and $\epsilon$ as:
\begin{equation}
U(1+\omega,\epsilon)=1+\dfrac{1}{2}...
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1answer
87 views
Why in quantum mechanics must orthogonal states stay orthogonal? [duplicate]
Given two states $|A(t)\rangle$ and $|B(t)\rangle$. If $\langle A(0)|B(0)\rangle=0$ then for all $t$, $\langle A(t)|B(t)\rangle=0$.
This is a fundamental rule of quantum mechanics. And we can imply ...
6
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2answers
421 views
Logarithm of Operators in Quantum Mechanics
In an operators algebra $\mathcal{A}$ one can consider a self-adjoint (i.e. real) operator $H$ and note that $$U=e^{iH}$$ exists and is unitary. A mathematical question will be whether any unitary ...
2
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1answer
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Why do we use infinitesimal forms of operators?
In many undergraduate texts on quantum mechanics (I'm using Modern Quantum Mechanics 2nd Edition by Sakurai as reference here), the treatment of angular momentum goes something along the lines of:
...
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Are basis vectors time-independent in the Schrödinger picture? If yes, are they time-dependent in the Heisenberg picture?
If I am not wrong, the Schrödinger Picture requires its basis vectors to be time-independent. My logic is such: The state vector $|\psi\rangle$ of the system will be in a superposition of the basis ...
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2d CFT and first intersection of vanishing curves
In the search for 2d Unitary CFT's, we use the Kac determinant to find the null curves, and remove regions in parameter space(c and h space) where the determinant is negative.
Now, for $h>0$ and $ ...
4
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1answer
254 views
Does Haags Theorem forbid Time-Evolution?
I didn't quite grasp the essence of Haags Theorem in the the way it is presented (for example on wikipedia), but the issue seems to be that if one wants to represent infinitely degrees of freedom ...
2
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0answers
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Feynman propagator for Dirac fields and $i\epsilon$ prescription for analytic continuation
The analytic continuation from Euclidean space to Minkowski spacetime is perturbatively well (uniquely) defined to all orders for the Feynman propagator for Dirac fields with the so called "$i\epsilon$...
2
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0answers
137 views
Optical theorem in QFT
I've been working with the Optical theorem in the case in which final and initial states are equals and I have the following doubt. Let's write the scattering matrix $S$ as:
$$S = 1 + i·T \tag1$$
...
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3answers
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Why is it important that physics consists of unitary phenomena? [duplicate]
AFAIK all known quantum laws are unitary (except the collapse postulate, which is dubious anyway).
Why is it important that the laws be unitary?
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What is the unit of polarization mode dispersion?
I calculated the differential group delay (DGD) from measured data following a chapter in the book "Polarization Optics in Telecommunication" by Jay Damask.
Basically, I measured a unitary matrix $U(...
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2answers
127 views
Is Quantum Mechanics Compatible with Conservation of Information?
What is exactly the law of conservation of information? In quantum mechanics we have truly random outcomes in experiments, but doesn't this randomness mean that new information is produced and the law ...
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2answers
44 views
Quickly write a unitary transformation connecting two arbitrary states
This is for an introductory QM course. Say, I have pure states $\vert\alpha\rangle$ and $\vert\beta\rangle$. I would like to find a unitary operator that takes me from one to the other. What is the ...
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3answers
256 views
What is meant by unitary time evolution?
According to the time evolution the system changes its state the with the passage of time. Is there any difference between time evolution and unitary time evolution?
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1answer
54 views
Unitary CFT and Kac determinant
I am currently reading the BPZ paper, and trying to understand the nature of Minimal models.
We compute the Kac determinant, and it is claimed that it must be positive for the theory to be a Unitary ...
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4answers
1k views
Why can't two different quantum states evolve into the same final state?
Is it true that two different states cannot evolve into the same final state? Can they achieve this state at different times? If yes, what is the proof?
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0answers
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Anti-commutative hermitian operators
I have some trouble to prove the next statements:
Let $A,B$ two anti-commutative hermitian operators, i.e. $\{A,B\}=AB+BA=0$. Does $A$ and $B$ share any eigenket?.
If $U$ is an unitary ...
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Are there any black hole information loss solutions that do not resort to non-locality?
It seems as though all solutions to the information paradox resort to non local effects. What solutions do we have that preserve locality?
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1answer
84 views
What is the physical significance/importance of anti-unitary operators?
Time-reversal symmetry is an anti-unitary operator.
I understand the mathematical definition of this, but what are the implications? What should/would one expect from anti-unitary operators? Are ...
3
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Use of Cutkosky rule, the Optical Theorem and Regge trajectories in pp scattering total cross-section calculation
Cutkosky rule states that:
$$2Im \big(A_{ab}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c p^{\mu}_{c}-\sum_a p^{\mu}_{a}\Big)|A_{cb}|^2\hspace{0.5cm} (1)$$
putting $a=b=p$ in Cutkosky rule we deduce the ...
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1answer
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Can someone explain why this matrix is unitary? [closed]
I have a matrix
$$
\begin{bmatrix}
a & b \\
e^{i\theta}b^* & -e^{i\theta}a^*
\end{bmatrix}
$$
Where $\theta$ is a real number, and $a$ and $b$ are complex number such that $|a|^2 + |...
2
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1answer
144 views
Unitary Representations in Conformal Field Theory
So I am currently studying conformal field theory from the perspective of the representation theory of Lie algebras. I am trying to understand exactly why we care about unitarizable Verma modules. For ...
2
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2answers
238 views
Two Level Atom Rotating Frame
Introduction
I am trying to work out the Rabi problem. In particular I am trying to work it out from a perspective focusing on the operators rather than the kets. Think Heisenberg picture rather than ...
3
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0answers
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Analytic cotinuation between Minkowskian and Euclidean space, and causality
We can flip between Minkowkian and Euclidean signature by Wick rotation, and it is a well defined operation, provided there are no non - trivial singularities.
Now, Unitarity in Minkowskian space ...
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54 views
If a theory violates unitary and renormalization of quantum theory, is it wrong?
If a theory of quantum gravity violates unitarity and renormalization at high energies, does that mean it is wrong and cannot be correct?
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1answer
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Quick Question About Conformal Blocks
When a conformal block has dimensions and spin that violates its unitary bounds, does that make the block equal to zero. I'm asking because I'm trying to calculate 3D conformal blocks via a recursion ...
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0answers
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Why can we use time-dependent perturbations when evaluating the S-matrix?
Suppose we have Hamiltonian $H_0 + V$. When working in the interaction picture we may derive the evolution operator of $|\psi_I(0)\rangle$ which is given by $$S(t,t_0) = T\left[\exp \left( -i \int_{...
3
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1answer
79 views
A question on the no-cloning theorem
When reading papers on the no-cloning theorem, I understand that one is searching for a unitary matrix $U$, such that for any state
$\vert \phi_A \rangle$ in a Hilbert state $H_A$, we have that
$U\...
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0answers
32 views
Can there be unitarily inequivalent representations in free QFT on a flat background?
In https://www.youtube.com/watch?v=9OGfvwt3H74 starting from 48:20 up to approximately 50:00, the person speaking says that different time slicing can lead to unitarily inequivalent representations ...
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What effect would result from always keeping Page time >> Age of universe?
A thought experiment: I imagine, similar to early theories (eg: Hoyle),
that the universe’s mass is not constant, but increases over cosmic
time, in such a way that even the thinnest vacuum will ...
3
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1answer
98 views
What is the most likely/favoured theory put forth explaining the Black hole information paradox?
What is the most likely/favoured theory put forth explaining the Black hole information paradox? Is there any theory that is most favoured that has no flaws, or most likely explains how black holes ...
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0answers
32 views
What is the problem of non-unitarity in RQM?
In Peskin and Schroeder, section 3.2, it is stated that Lorentz group being non-compact it does not have any finite dimensional, faithful unitary representation. But it has also been said that one ...
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0answers
54 views
Non-unitarity of finite dimensional Lorentz group and its implications
In Peskin and Schroeder, section 3.2, it is stated that Lorentz group being non-compact it does not have any finite dimensional, faithful unitary representation. But it has also been said that one ...
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3answers
301 views
How to find unitary matrices?
I'm having trouble fully wrapping my head around unitary matrices. I'm working on them in relation to quantum mechanics. The question specifically I am working on is:
Given the Pauli matrices $\...
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1answer
71 views
Is *Conservation of Distinction* a true conservation law in mainstream physics?
Both Leonard Susskind and Francis Heylighen have written about the Conservation of Distinction but it seems Susskind more closely connects this (law?) with unitarity in quantum mechanics. Heylighen ...
2
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1answer
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Equivalence between ghosts?
Ok. I'm trying to get the terminology right about the term ghost in physics. Is there any equivalence between these terms?
Faddeev-Popov ghosts
Paul-Villars ghosts
Landau ghost
The vanishing ...
4
votes
1answer
116 views
Cutkofsky cutting rules questions
Consider the two photon exchange diagrams above. The Cutkofsky rules tell us that the imaginary part of each of these diagrams can be obtained by cutting through the graphs in all possible ways such ...
1
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1answer
97 views
Wrong sign for the Laplacian in Schrodinger equation?
Suppose after some calculations I arrive at an equation of the form:
\begin{equation}
[\nabla^2 +V(\mathbf{r})]\psi(\mathbf{r})=k^2\psi(\mathbf{r}) \tag{1}
\end{equation}
As you can see here the ...
12
votes
2answers
389 views
Why does unitarity require the Higgs to exist?
A standard argument that the Higgs boson must exist is that without it, amplitudes in the Standard Model at the TeV scale violate unitarity. This is explained in section 21.2 of Peskin and Schroeder ...
