# Why unitary evolution?

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Why is the fundamental evolution of systems in non-relativistic quantum theory postulated to be unitary rather than a more general CPTP map?
The usual justification for why the evolution of physical systems is unitary in quantum mechanics involves arguments like "probabilities must sum to 1" or similar arguments that apply equally to any CPTP map. I'm just curious what justifications people here would use for selecting out unitary evolution in particular.

To be clear this is really for gathering ideas on teaching non-relativistic QM. I'm not disputing unitary evolution or anything like that and am aware of some justifications for postulating fundamental unitary evolution. By necessity I am assuming responses are familiar with CPTP maps.

Edit: Though happy to hear about why the evolution might be non-unitary fundamentally

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bhobba, Greg Bernhardt and dextercioby

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CPTP?

Demystifier
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CPTP?
Completely Positive Trace Preserving.

vanhees71 and hutchphd
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Completely positive trace-preserving maps. Applies to generalizations of density operators.

bhobba, topsquark, vanhees71 and 2 others
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TL;DR Summary: Why is the fundamental evolution of systems postulated to be unitary rather than a more general CPTP map?

The usual justification for why the evolution of physical systems is unitary in quantum mechanics involves arguments like "probabilities must sum to 1" or similar arguments that apply equally to any CPTP map. I'm just curious what justifications people here would use for selecting out unitary evolution in particular.

To be clear this is really for gathering ideas on teaching. I'm not disputing unitary evolution or anything like that and am aware of some justifications for postulating fundamental unitary evolution. By necessity I am assuming responses are familiar with CPTP maps.
Such subtleties are usually answered by the "symmetry approach". I must admit, the more I think about, how to teach introductory quantum mechanics the more desparate I get, because you cannot use these arguments, which involves the theory of Lie groups and Lie algebras as well as their ray representations on a Hilbert space, which are pretty advanced mathematical concepts. Of course you cannot in any way derive quantum mechanics from classical mechanics in a deductive way, because quantum mechanics is by far the theory with the larger realm of applicability, and classical mechanics is an approximate effective description for much coarse-grained macroscopic observables of many-body systems, but that's also a can of worms of course.

For me the most plausible heuristics is, assuming you can argue with group theory, that you look for a quantum theory thats dynamics is compatible with the symmetries of Newtonian spacetime, and it needs to be only the part of the corresponding Galilei group that is smoothly connected with the identity. From the general framework of QT this implies that you look for unitary ray representations of the Galilei group, and that's most easily done by first looking for such representations for the corresponding Lie algebra.

Then it turns out that the unitary ray representations that lead to a satisfacoritly interpretable quantum dynamics can be lifted to a unitary representation of an extended Lie algebra, which substitutes the rotation subgroup by SU(2) (its covering group) and has the one non-trivial central charge ##\neq 0##, and this central charge represents the mass of the system.

The by definition the time evolution is realized by the time translations with the Hamiltonian as the generator, and since we have lifted the ray representations without loss of generality to unitary representations of the (somewhat extended quantum) Galilei group, the time evolution is a unitary transformation. You can also show that you can define a position observable in the proper sense.

You find a very good treatment of the representation theory of the Galilei algebra (group) in Ballentine, Quantum Mechanics.

Of course, this approach cannot be used as an introductory framework for the QM 1 lecture. Here I have no better idea to start with a historical approach, emphasizing the failure of Bohr-Sommerfeld quantization, and then use the "wave-particle dualism" of "old quantum mechanics" (emphasizing from the beginning that this is fact is a dead end and one of the reasons why modern QM has been worked out pretty soon!) to motivate the Schrödinger equation as a wave equation describing de Broglies "matter waves", which of course from the very beginning have to be interpreted in the probabilistic way a la Born. From this you can motivate the Hilbert-space formulation with the representation of the observables by self-adjoint operators.

Then you can also use the Diracian approach of "canonical quantization", assuming you students have heard about Hamiltonian analytical mechanics and Poisson brackets in the mechanics lecture. Then for the classical point particle you simply start with ##\{\cdot,\cdot \} \rightarrow [\cdot,\cdot]/\mathrm{i} \hbar## and the "Heisenberg algebra" for position and momentum (for the beginning motion in 1 dimension is enough) with ##[\hat{x},\hat{p}]=\mathrm{i} \hbar \hat{1}##. From this you can construct position and momentum eigenstates by using the fact that ##\hat{p}## is the generator for spatial translations, i.e., it has the same mathematical meaning as in classical mechanics, where ##p## also is the generating function for an infinitesimal canonical transformation, describing spatial translations.

bhobba, PhDeezNutz, Maarten Havinga and 2 others
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For me the most plausible heuristics is, assuming you can argue with group theory, that you look for a quantum theory thats dynamics is compatible with the symmetries of Newtonian spacetime, and it needs to be only the part of the corresponding Galilei group that is smoothly connected with the identity. From the general framework of QT this implies that you look for unitary ray representations of the Galilei group, and that's most easily done by first looking for such representations for the corresponding Lie algebra
Thank you, this is a very nice argument. General CPTP maps are not compatible with the symmetries of Newtonian spacetime. 👍

bhobba
Maarten Havinga
Might this also be related to energy conversation? For some reason I have a feeling energy wouldn't be conserved in non unitary evolution, due to eigenvalues of norm less or greater than 1.

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Thank you, this is a very nice argument. General CPTP maps are not compatible with the symmetries of Newtonian spacetime. 👍
While that is true, it must be said that „Newtonian spacetimes” are substituted for curved spacetimes in GR or spacetimes with curvature and torsion in the Einstein-Cartan version, so that any extension of the known formalism of the PVM/POVM approach to quantum theory is well justified, so I see no reason NOT to go to CPTP maps. Indeed, the „perfect mathematics” which comes from unitarity (I mean all the beautiful functional analytic + Lie groups/algebras theorems) + „unitarity preserves probabilities” seem an ideal combination, but I think it cannot be the whole story. Pinging prof. Arnold Neumaier. I will like to see what he says to you. @A. Neumaier

bhobba, gentzen, vanhees71 and 1 other person
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While that is true, it must be said that „Newtonian spacetimes” are substituted for curved spacetimes in GR or spacetimes with curvature and torsion in the Einstein-Cartan version
That's certainly true. In fact there's fairly good arguments that given our cosmology is deSitter and seems to have increasing degrees of freedom due to cosmological acceleration that the evolution cannot be unitary fundamentally. My apologies I'll edit the opening post to say this is more to do with non-rel QM and teaching rather than "truly" fundamental.

Although I don't mind reading what people have to say about the curved spacetime case since it's a natural continuation of the discussion.

bhobba and vanhees71
Thank you, this is a very nice argument. General CPTP maps are not compatible with the symmetries of Newtonian spacetime. 👍
??? Why not?

I'm just curious what justifications people here would use for selecting out unitary evolution in particular.
The reason unitarity is thought to be fundamental is because
• the unitarity of the S-matrix has lots of useful experimental consequences, and
• the better a system is isolated the less dissipation is found experimentally.
It is therefore very natural to assume that a completely isolated system behaves unitarily.

The completely positive dynamics emerges from the unitary evolution by projecting out the environment and usining the Markov approximation. The latter is valid whenever all relevant degrees of freedom that could act as a memory have been taken into account.

Thus there is little incentive to take copmletely positive maps as fundamental.

bhobba, Couchyam, LittleSchwinger and 1 other person
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??? Why not?
For example that they aren't time translation and time inversion symmetric. Perhaps I'm wrong.

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While that is true, it must be said that „Newtonian spacetimes” are substituted for curved spacetimes in GR or spacetimes with curvature and torsion in the Einstein-Cartan version, so that any extension of the known formalism of the PVM/POVM approach to quantum theory is well justified, so I see no reason NOT to go to CPTP maps. Indeed, the „perfect mathematics” which comes from unitarity (I mean all the beautiful functional analytic + Lie groups/algebras theorems) + „unitarity preserves probabilities” seem an ideal combination, but I think it cannot be the whole story. Pinging prof. Arnold Neumaier. I will like to see what he says to you. @A. Neumaier
Energy conservation cannot be an argument though, because also for explicitly time-dependent Hamiltonians, where energy conservation doesn't hold, because then ##\mathring{\hat{H}}=\partial_t \hat{H} \neq 0##, the time evolution is unitary since ##\hat{H}## is still self-adjoint at all times. The time evolution operator then is formaly given as
$$\hat{U}(t,t_0)= \mathcal{T}_c \exp \left [-\mathrm{i} \int_{t_0}^{t} \mathrm{d} t' \hat{H}(t') \right],$$
where ##\mathcal{T}_c## is the "causal time-ordering opertor", i.e., ordering the ##\hat{H}(t_k)## in the Taylor-series expansion of the exp function with increasing time arguments from right to left.

bhobba, Maarten Havinga and LittleSchwinger
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The completely positive dynamics emerges from the unitary evolution by projecting out the environment and usining the Markov approximation.
Sorry I just read over this. Aren't CPTP maps just formed by reduction of unitary evolution to a subsystem (Stinespring's theorem), I don't recall seeing Markov approximations mentioned.

bhobba and vanhees71
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The more I think about open quantum systems I come to the conclusion that enforcing Markov approximations is evil. The most systematic approach is the Lindblad equation, but nobody tells me, how to correctly choose the Lindblad operators and it's not clear to me, how to ensure the correct equilibration to Bose-Einstein or Fermi-Dirac (or at least Boltzmann, if it's an appropriate approximation) statistics.

bhobba and LittleSchwinger
For example that they aren't time translation and time inversion symmetric. Perhaps I'm wrong.
They can be translation invariant. That they are not invariant under time reversal is nowhere a requirement.
Sorry I just read over this. Aren't CPTP maps just formed by reduction of unitary evolution to a subsystem (Stinespring's theorem), I don't recall seeing Markov approximations mentioned.
Stinespring's theorem gives a single CP map from a single unitary map. But to get CP evolution from unitary evolution the Markov approximation is needed.

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bhobba, LittleSchwinger and vanhees71
The more I think about open quantum systems I come to the conclusion that enforcing Markov approximations is evil. The most systematic approach is the Lindblad equation,
It is heavily used in applications, so it cannot be all that evil....

More heavy non-Markovian machinery is needed only for multiscale applications (two different time scales to be modelled, or very long time behavior).

but nobody tells me, how to correctly choose the Lindblad operators
They are chosen according to experience from what are the likely processes that matter - not too unsimilar to how one chooses effective actions in QFT.

and it's not clear to me, how to ensure the correct equilibration to Bose-Einstein or Fermi-Dirac (or at least Boltzmann, if it's an appropriate approximation) statistics.
The latter is given by enforcing fluctuation-dissipation relations.

bhobba, LittleSchwinger, gentzen and 1 other person
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That they are not invariant under time symmetry is nowhere a requirement.
Sorry just to be clear, do you mean "not invariant under time inversion symmetry"?

And if so, why would you say this isn't a requirement?

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Time-reversal symmetry is not a symmetry of Nature, and what should this "discrete symmetry" have anything to do with (continuous) time evolution?

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Time-reversal symmetry is not a symmetry of Nature, and what should this "discrete symmetry" have anything to do with (continuous) time evolution?
Thanks that's sort of what I'd say.

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Thanks in general, I think I have some good material now.

bhobba and vanhees71
Maarten Havinga
Energy conservation cannot be an argument though, because also for explicitly time-dependent Hamiltonians, where energy conservation doesn't hold
Point taken.

From wikipedia:
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product.
So it preserves the inner product aka angles, not just norm (the trace). Thus I think this is specifically required for the conservation of angular momentum.

Edit: angles between states of course, not between vectors... so I've looked it up, and the square of an inner product of pure states is called the fidelity. It's a measure of how likely you confuse the two states with each other. The fidelity between states is conserved, apparently.

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LittleSchwinger and vanhees71
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So it preserves the inner product aka angles, not just norm (the trace). Thus I think this is specifically required for the conservation of angular momentum.
There's actually a stronger result called Uhlhorn's theorem that proves that if time evolution preserves the orthogonality of states, so:
##(\phi,\psi) = 0##
implies
##(\phi(t), \psi (t)) = 0##
Then the evolution operator must be unitary. This is much stronger than Wigner's theorem which proves the same assuming the angles are constant for all states.

This orthogonality preservation is required for other symmetries as you say and is the main reason I would rule out CPTP evolutions as fundamental in non-relativistic QM.

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bhobba and vanhees71
andresB
There's actually a stronger result called Uhlhorn's theorem that proves that if time evolution preserves the orthogonality of states, so:
##(\phi,\psi) = 0##
implies
##(\phi(t), \psi (t)) = 0##
Then the evolution operator must be unitary.

Apparently, there are even stronger results that say that if time evolution preserves any angle (Not just ##\frac{\pi }{2}##) then the evolution operator must be unitary

bhobba, DrClaude, gentzen and 3 others
Couchyam
Are we trying to compile empirical evidence of unitarity (i.e. explain how we "know" that time evolution is in fact unitary) or suggest philosophical(?) reasons for why time evolution is unitary?
Also, are we focusing exclusively on the dynamics of processes that don’t contribute to the arrow of time?
(For the record, I think the evidence for unitarity is more suggestive than definitive, and comes from the fact that there is no evidence of violations of unitarity in high energy/short timescale physics, which presumably set the tone for everything else; a more general/predictive theory of relativistic QM might model wave function collapse as a unitary process, and we just haven’t stumbled into the right ‘perspective’ on it yet.)

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A (somewhat indirect) evidence for the unitary of time-evolution is the unitarity of the S-matrix for scattering processes. From the unitarity of the S-matrix in quantum-many-body theory you can derive, via the Kadanoff-Baym equations, coarse-grained to the Boltzmann-(Uehling-Uhlenbeck) transport equations, the Boltzmann (Bose-Einstein, or Fermi-Dirac) statistics of equilibrium (detailed-balance relations of the S-matrix elements and the H-theorem). In this sense these (quantum) thermal distributions are evidence for the validity of the unitarity of the S-matrix. Note that, despite for claims often to be read in textbooks, there's no need for parity or time-reversal symmetry for the detailed-balance relations to be valid.

bhobba, dextercioby and LittleSchwinger
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Are we trying to compile empirical evidence of unitarity (i.e. explain how we "know" that time evolution is in fact unitary) or suggest philosophical(?) reasons for why time evolution is unitary?
I would say we want empirical evidence. The non-empirical part will enter anyway during weighting and interpretation of the evidence: Why is the evidence convincing and relevant?

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The only thing that's convincing about a theory is its ability to describe accurately the empirical phenomena. That's why string theory et al are not very convincing for me ;-).

gentzen
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Are we trying to compile empirical evidence of unitarity (i.e. explain how we "know" that time evolution is in fact unitary) or suggest philosophical(?) reasons for why time evolution is unitary?
It's just for a class in non-Rel QM, nothing more. I'm aware you can have well-defined non-unitary evolution at a fundamental level in for example algebraic treatments of non-perturbative QED. However that's well beyond what I'm interested in here.

bhobba and vanhees71
Couchyam
It's just for a class in non-Rel QM, nothing more. I'm aware you can have well-defined non-unitary evolution at a fundamental level in for example algebraic treatments of non-perturbative QED. However that's well beyond what I'm interested in here.
Also (maybe I should have asked this earlier) are you interested primarily in unitarity of observable quantum dynamics (i.e. of measurable wave functions) or the possibility of unitarity in an absolute, all-encompassing sense (i.e. of the whole universe, including measurement apparatus/observers?)

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Also (maybe I should have asked this earlier) are you interested primarily in unitarity of observable quantum dynamics (i.e. of measurable wave functions) or the possibility of unitarity in an absolute, all-encompassing sense (i.e. of the whole universe, including measurement apparatus/observers?)
The former. It's a typical enough non-Rel QM course, so not treating macroscopic bodies in detail. Certainly not the whole universe! You couldn't even approach dealing with cosmology in non-Rel QM.

bhobba
Couchyam
@LittleSchwinger: Also, are you looking for a universality argument (i.e. something like 'unitarity is to be expected'), or explanations and didactic examples that would help build physical intuition?

bhobba and vanhees71
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Probably more the former, but it doesn't matter too much. It's just that the usual argument of "probabilities must sum to one" isn't really sufficient.

bhobba and vanhees71
Couchyam
Probably more the former, but it doesn't matter too much. It's just that the usual argument of "probabilities must sum to one" isn't really sufficient.
I thought the usual argument was that unitarity is to wave function (or Hilbert space?) dynamics what time reversibility (or maybe invertibility, and maybe also Liouville's theorem for numerical stability) is to classical dynamics: even if the laws of nature weren't invariant under time reversal, one could at least in principle infer the past (or future) of a wave function to arbitrary precision from its present value. One could concoct non-unitary maps that are reversible or invertible and that don't 'compress' Hilbert space weirdly and irreversibly, but those maps would probably inevitably break continuity (or violate the 'no distortion' condition.) That being said, wave function collapse is understood to be a relatively discontinuous process, and is certainly non-unitary, and so it could be that operations/transformations of comparable abruptness occur at comparably (in)frequent intervals. It might be interesting to look at ensembles of weak non-unitary transformations (small or infinitesimal deviations from the identity) and how they collectively influence the wave function: for example, would their combined effect result in a more severe non-unitary transformation, or would their non-unitarity somehow 'cancel out' so that they could be approximated by a unitary map? You might also consider the induced dynamics of a wave function with a trivial Hamiltonian (i.e. proportional to the identity) that is measured periodically or intermittently (at regular or random intervals) through observables that undergo a prescribed dynamics.

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I thought the usual argument was that unitarity is to wave function (or Hilbert space?) dynamics what time reversibility (or maybe invertibility, and maybe also Liouville's theorem for numerical stability) is to classical dynamics: even if the laws of nature weren't invariant under time reversal, one could at least in principle infer the past (or future) of a wave function to arbitrary precision from its present value.
Time reflection symmetry has nothing to do with the unitary time evolution. In the parts of Nature (i.e., neglecting the weak interaction) it's an additional discrete symmetry, which must necessarily be represented as an anti-unitary transformation since the Hamiltonian must stay bounded from below under time-reversal. Although with the weak interaction the time evolution in Q(F)T is unitary.

The symmetry argument for unitarity of the time evolution is that time-translation invariance as a continuous symmetry must be realized as unitary transformation.

bhobba
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The symmetry argument for unitarity of the time evolution is that time-translation invariance as a continuous symmetry must be realized as unitary transformation.
This is what makes unitarity less fundamental in QFT in curved spacetime, since there we do not have time-translation invariance.

The symmetry argument for unitarity of the time evolution is that time-translation invariance as a continuous symmetry must be realized as unitary transformation.
Coming back to non-Rel QM, this is basically what I was asking. Since one can have a CPTP evolution, why wouldn't you. A. Neumaier's argument that isolated systems do seem to undergo unitary evolution empirically is perfectly correct, but I was wondering if there was a more "first principles" reason.

bhobba and vanhees71