48 votes
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Why is the information paradox restricted to black holes?

(The answers by Mark H and B.fox were posted while this one was being written. This answer says the same thing in different words, but I went ahead and posted it anyway because sometimes saying the ...
Chiral Anomaly's user avatar
31 votes

How can the universe evolve unitarily if there's no clock outside it?

You are assuming that time does not exist without clocks. That is analogous to assuming that space does not exist without rulers, and both assumptions are unjustified. As far as we know, we live in a ...
Marco Ocram's user avatar
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23 votes
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Where does the $i$ come from in the Schrödinger equation?

Let $U$ be an unitary operator. Write $$ U=\mathbb I+\epsilon A $$ for some $\epsilon\in\mathbb C$, and some operator $A$. Unitarity means $U^\dagger U=\mathbb I$, i.e., $$ U^\dagger U=(\mathbb I+\...
AccidentalFourierTransform's user avatar
22 votes

Why is the information paradox restricted to black holes?

When Dr. Hawking talks about information being destroyed, he is talking about the erasure of all evidence that the information ever existed. In the case of burning a written letter, you could track ...
Mark H's user avatar
  • 24k
21 votes

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? That depends on exactly what you mean. If we consider the total state of a closed system, then two different states will ...
DanielSank's user avatar
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18 votes
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QED and anomaly

1. How can we show that $\partial\cdot j\equiv 0$ at the quantum level? For example, by showing that the Ward Identity holds. It should be more or less clear that the WI holds if and only if $\partial\...
AccidentalFourierTransform's user avatar
18 votes

Are all representations of a finite group unitary?

Every representation $(D,V)$ of a finite group $G$ is equivalent to a unitary representation. It is often termed as Weyl's unitary trick. This works by simply redefining your inner product by ...
exp ikx's user avatar
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16 votes
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Why do negative norm states break unitarity?

I asked Mark Srednicki about this, and he told me that it's not really correct to say that negative-norm states break unitarity, because negative-norm states don't exist by the definition of the inner ...
tparker's user avatar
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16 votes
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On finite-dimensional unitary representations of non-compact Lie groups

Proposition. Let $G$ be a connected non-compact Lie group that is a semisimple Lie group and $$U: G \ni g \mapsto U_g \in B(H)$$ ($B(H)$ being the set of bounded operators $A:H \to H$) a continuous ...
Valter Moretti's user avatar
16 votes

What is meant by unitary time evolution?

Yes, there is a difference. Unitary time evolution is the specific type of time evolution where probability is conserved. In quantum mechanics, one typically deals with unitary time evolution. ...
Avantgarde's user avatar
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15 votes
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How can the universe evolve unitarily if there's no clock outside it?

I'm not sure an answer exists to your question because it is not stated sufficiently precisely. However I think it is worth pointing out that we need to distinguish between the time coordinate and the ...
John Rennie's user avatar
13 votes
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Can we make sense of a Hamiltonian $a^\dagger a^\dagger + a a$?

No, the ground state is not well-defined because the energy is unbounded below. To see this, switch back to the variables $x$ and $p$ using $a \sim x + ip$ to find $$H \sim p^2 - x^2.$$ This is the ...
knzhou's user avatar
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12 votes
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What does sandwiching with an unitary operator and its inverse imply?

This goes back to the paradigm that if a ket transforms as $|\psi\rangle\to \hat{U}|\psi\rangle$ under a unitary symmetry transformation $\hat{U}$, then a bra transforms as $\langle\phi|\to \langle\...
Qmechanic's user avatar
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11 votes
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Symmetry transformations on a quantum system; Definitions

There are many definitions of symmetry in Quantum Theory. However your idea is not correct: symmetries, acting on states, do change outcomes of measurements, at least for one observable. Otherwise we ...
Valter Moretti's user avatar
11 votes

Logarithm of Operators in Quantum Mechanics

The Stone's theorem proves the following. Consider a group of unitary operators $(U(t))_{t\in\mathbb{R}}$ acting on a Hilbert space $\mathscr{H}$ (i.e. satisfying $U(t+s)=U(t)U(s)$, in more ...
yuggib's user avatar
  • 12k
11 votes
Accepted

Are all representations of a finite group unitary?

They are all equivalent to unitary representations. It's not quite the same thing. Merely checking unitarity might not do the trick. For instance \begin{align} \Gamma(e)&=\left(\begin{array}{...
ZeroTheHero's user avatar
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11 votes
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What does it mean for QFT to be unitary?

Maybe this is a boring answer but unitarity of a QFT means states at different times can be related to each other by \begin{equation} \left | \Psi(t_2) \right > = U \left | \Psi(t_1) \right > \...
Connor Behan's user avatar
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11 votes
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Is unitary time evolution the same as obeying the Schrödinger equation?

As other answers have pointed out, assuming unitary time evolution does get you part of the way there — but not all the way. In particular, you could call it a postulate that the Hermitian operator ...
Jbag1212's user avatar
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10 votes
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Why do quantum gates have to be reversible?

Quantum gates have to be reversible because quantum mechanics is reversible (and even more specifically it is unitary). It's just an observed fact about the universe. (Even measurement can be modeled ...
Craig Gidney's user avatar
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10 votes

Allowed Field Re-definitions in QFT

After the field redefinition $$ \phi \rightarrow \phi + \frac{\partial_{\nu}\partial^{\nu}}{v^2}\phi\tag{1} $$ there is an apparent violation of unitarity because the propagator decays too fast in ...
AccidentalFourierTransform's user avatar
10 votes
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Why are symmetry transformations connected to the identity necessarily represented by linear unitary operators?

Well, the point is not continuity of the representation (if continuity is referred to the representation, see the final comment), but a related deeper fact concerning the group itself (though some ...
Valter Moretti's user avatar
10 votes

Why is the information paradox restricted to black holes?

I must admit, I'm not the most qualified person to answer here. If you place the burning letter into a sealed box (a closed system) and allow the system to carry itself to its final state, with a ...
BenjaminF's user avatar
  • 249
10 votes
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What is so problematic in the information paradox?

Quantum mechanical evolution via the Schrodinger equation (or whatever your favorite formalism is) is unitary. We can start with a pure state with some particles and no black holes, and throw these ...
Andrew's user avatar
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10 votes
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Why is imaginary time evolution non-unitary?

If $H$ is hermitian then $U=e^{-itH}$ is unitary if and only if $t$ is real. Making a change of variables $t=i\tau$ won't change that. The point is that when you do a Wick rotation to imaginary time ...
jacob1729's user avatar
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9 votes
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Griffith's Proof that a wave function will stay normalized is incorrect?

We know, from Schrodinger's equation, that $$ \frac{d\psi}{dt}=\left(\frac{i\hbar}{2m}\frac{d^2}{dx^2}-\frac{i}{\hbar}V(x)\right)\psi $$ Taking the conjugate tells us how $\psi^*$ evolves. $$ \frac{...
Jahan Claes's user avatar
  • 8,100
9 votes
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Why transformations in quantum mechanics are linear?

Short answer since I'm on mobile: No, it's not a postulate but rather a theorem. First, clear things up a little. We want a symmetry of the theory to act as an arbitrary transformation which ...
Moe's user avatar
  • 503
9 votes
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Why is time-evolution operator unitary?

1st point of view: If you accept the Schrödinger equation $$ \mathrm i\hbar\, \partial_t \psi = \hat H \psi $$ with self-adjoint $\hat H$, then your equation 1 follows directly and $\hat U$ is ...
Noiralef's user avatar
  • 7,238
9 votes
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Why is quantum mechanics reversible?

So before college even, we students learn that there is this wonderful alternate way to look at a lot of physical problems in terms of energies. But, they seem to lose some information about the ...
CR Drost's user avatar
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9 votes
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Does Haags Theorem forbid Time-Evolution?

The answer is no, Haag's theorem does not prevent time-evolution in the Heisenberg picture. Given any representation and any unitary transformation, applying the latter to the former gives a ...
Chiral Anomaly's user avatar
9 votes

The structure of the Hilbert space of 2d CFT

It's no accident that you invoked ghosts to find a counter-example. Ghosts are non-unitary and the standard proof of the primary / descendant classification uses unitarity. In particular, given a ...
Connor Behan's user avatar
  • 7,236

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