Exact symmetry, quantum states, and symmetric dynamics

[jj90]

Hello everyone,

I am trying to verify the physical coherence, within standard quantum mechanics and relativistic quantum field theory, of an idealized but symmetry-constrained setup involving two arbitrarily complex macroscopic subsystems. The goal is to check internal consistency with accepted physics, independent of practical realizability.

Setup (idealized):
Consider a closed quantum system composed of two subsystems, A and B, arranged such that the entire global quantum state is exactly invariant under a 180° spatial rotation about a fixed axis. Equivalently, the global state and Hamiltonian commute with the corresponding rotation operator. The subsystems may be arbitrarily complex (macroscopic, many degrees of freedom). Importantly, symmetry is imposed at the level of the quantum state or observable algebra, not via classical particle trajectories or sharply localized configurations.

Questions:

1. State preparation
   Is it coherent within standard QM (and relativistic QFT) to posit such a globally symmetric quantum state—even for highly complex macroscopic subsystems—without assuming classical micro-determinacy of particle positions or trajectories? More generally, is exact spatial symmetry well-defined at the level of the quantum state or operator algebra, regardless of system complexity?
2. Symmetry vs uncertainty
   Does the Heisenberg uncertainty principle (or related quantum limitations) obstruct the existence of exact spatial symmetry at the level of the quantum state, or is symmetry properly understood as invariance under symmetry operators rather than as a claim about sharply defined particle configurations?
3. No-cloning considerations
   Does the quantum no-cloning theorem impose any principled restriction on preparing two subsystems in identical macroscopic quantum states, provided the states are mixed, known, or prepared symmetrically from the outset via a common dynamical process, rather than produced by cloning an unknown pure state?
4. Symmetry-preserving dynamics
   Suppose an idealized external interaction (a “machine”) acts on the system via unitary evolution that itself commutes with the same 180° rotation operator. Is it correct that such symmetry-preserving unitary dynamics necessarily maintain the global symmetry of the state, even if they involve highly nontrivial distortions, redistributions, or rearrangements of the subsystems’ degrees of freedom?
5. Exact vs FAPP symmetry
   In modern quantum theory, is there a principled distinction between a state being exactly symmetric (e.g., commuting with the symmetry operator) and being symmetric only “for all practical purposes” (FAPP)? Or is exact symmetry fully meaningful and well-defined at the level of the quantum state, independent of experimental accessibility or decoherence?
6. Decoherence and symmetry
   If decoherence or environment-induced superselection is invoked to explain emergent classical behavior in macroscopic systems, does such decoherence necessarily require an asymmetric environment? Or can a perfectly symmetric global quantum state remain exactly symmetric while still supporting locally classical, effectively distinguishable subsystems?

Scope clarification:
I am not asking about experimental feasibility, observers, or measurement outcomes, nor about interpretational commitments beyond standard quantum mechanics. The aim is simply to determine whether the assumptions above are internally coherent within accepted physical theory.

Any insight or references addressing these points would be very helpful.

 

[gentzen]

6. Decoherence and symmetry
If decoherence or environment-induced superselection is invoked to explain emergent classical behavior in macroscopic systems, does such decoherence necessarily require an asymmetric environment? Or can a perfectly symmetric global quantum state remain exactly symmetric while still supporting locally classical, effectively distinguishable subsystems?

A frequently used model has an effectively asymmetric environment:

In applications, a frequently used model universe consists of a tensor product of the system algebra $\mathbb{L}^S$ of the physical system under consideration and an environmental algebra $\mathbb{L}^E$ modeling the remainder of the universe by a heat bath with given temperature $T$.

No idea whether you could come up with a model with an environment respecting your 180° spatial rotation about a fixed axis.

The points 1. - 4. seem unproblematic to me. Point 5. should be unproblematic too, depending on how you interpret those concepts.

 

[jj90]

Thanks very much. That’s helpful, especially the confirmation that points (1)–(4) (and likely (5)) are unproblematic in principle.

I have a few brief clarifications, mainly to distinguish issues of modeling convenience from principled constraints in QM/QFT:

(1) Decoherence and symmetry:
Am I right in understanding that the asymmetry of the environment in standard decoherence models reflects a common modeling choice (e.g., heat baths, laboratories, etc.), rather than a fundamental requirement of quantum mechanics itself? In other words, is there anything in standard QM or relativistic QFT that forbids a globally closed system from remaining exactly symmetric at the level of the total quantum state, even if reduced subsystems exhibit effectively classical behavior?


(2) Symmetry vs uncertainty:
To be explicit, is it correct that exact spatial symmetry in QM/QFT is defined at the level of the quantum state and operator algebra (i.e., invariance under the relevant symmetry operator), rather than requiring sharply defined particle positions or classical trajectories? In that sense, does the uncertainty principle pose no obstacle to exact global symmetry of the state?


(3) Symmetry-preserving dynamics:
If a Hamiltonian or unitary evolution commutes with a given symmetry operator, is it correct that such symmetry-preserving dynamics cannot dynamically break that symmetry without introducing additional asymmetry into the dynamics itself? In particular, would a continuous family of symmetry-preserving unitaries preserve the symmetry throughout the evolution?


(4) Global vs perspectival symmetry breaking (optional clarification):
Would it be fair to say that any apparent symmetry breaking in such scenarios would be perspectival or relative to subsystems (e.g., reduced states), rather than an objective breaking of symmetry in the global quantum state?


Thanks again - this is exactly the kind of in-principle clarification I was looking for.

 

[QuarkyMeson]

I'm like 95% sure this is AI slop. Anyway:

Picture the whole universe as one perfectly reversible machine. Its state changes in a way that, in principle, you could run backward and recover exactly what it was before. A “symmetry” just means the machine doesn’t care if you relabel things in some way, like shifting every position by the same amount or rotating everything, the underlying rules don’t change under that relabeling. If the rules have that symmetry, then the universe can’t spontaneously start violating it just by running. If it starts off symmetric in that sense, it stays symmetric.

Now put a person inside the machine. The person can only look at a tiny part of the universe and ignores the rest. Because that small part is constantly interacting with everything else, information about delicate quantum “phase” relationships gets spread out into correlations with the rest of the universe. From the inside, once you ignore all those other details, the small part you track starts to look like it has settled into ordinary alternatives with no visible interference between them. That is decoherence. Nothing “happened to the universe’s state” in the sense of the whole machine becoming irreversible, what changed is that the interference became inaccessible when you focus on only a small piece.

This is closer to Everettian view of QM, which is as valid an interpretation as any. So that should answer what I think you're trying to ask.

 

[Nugatory]

In other words, is there anything in standard QM or relativistic QFT that forbids a globally closed system from remaining exactly symmetric at the level of the total quantum state, even if reduced subsystems exhibit effectively classical behavior?

There is not. Nothing stops you from writing down a wave function and a Hamiltonian with spherical symmetry. These might not describe any physically realizable system, but you’ve already excluded those concerns: "I am not asking about experimental feasibility, observers, or measurement outcomes".

is it correct that exact spatial symmetry in QM/QFT is defined at the level of the quantum state and operator algebra (i.e., invariance under the relevant symmetry operator), rather than requiring sharply defined particle positions or classical trajectories?

Yes. Particle positions and trajectories are something that we calculate from the quantum state, not something that we use to specify the quantum state.

In that sense, does the uncertainty principle pose no obstacle to exact global symmetry of the state?

The uncertainty principle doesn’t stop us from writing down any quantum state we please, whether symmetric or not. It tells us something about the statistical distribution of position and momentum measurements ia large number of systems prepared in whatever state we’re considering.

If a Hamiltonian or unitary evolution commutes with a given symmetry operator, is it correct that such symmetry-preserving dynamics cannot dynamically break that symmetry without introducing additional asymmetry into the dynamics itself? In particular, would a continuous family of symmetry-preserving unitaries preserve the symmetry throughout the evolution?

Yes, but that’s just doing math with abstract mathematical objects that don’t necessarily correspond to any realizable physical syste. You’ve written down a symmetrical state, you’ve applied operators that don’t affect the symmetry, of course the result is symmetrical.

Would it be fair to say that any apparent symmetry breaking in such scenarios would be perspectival or relative to subsystems (e.g., reduced states), rather than an objective breaking of symmetry in the global quantum state.

Under the conditions you started with there never would be any symmetry breaking. If any asymmetry is observed, it just tells us that the setup (initial state, Hamiltonian, measurements) isn’t completely symmetric - and that’s not surprising because it was an unrealizable idealization from the start.

 

[gentzen]

Now put a person inside the machine. The person can only look at a tiny part of the universe and ignores the rest. Because that small part is constantly interacting with everything else, information about delicate quantum “phase” relationships gets spread out into correlations with the rest of the universe.

So the small part broke the symmetry, no? And not just for itself, but globally! Or maybe not, if we assume that another indistinguishable small part also exist, as required by the symmetry under that 180° spatial rotation about a fixed axis.
The interesting question is what "indistinguishable" means or implies in this context. In the end, it probably means that we are allowed to take the quotient by the symmetry, and end up in a situation where only one small part is left, and the description is reduced.

I'm like 95% sure this is AI slop.

I have the impression that jj90 genuinely wants to understand whether the symmetry involving his "two arbitrarily complex macroscopic subsystems" would break, and whether one the given six quantum-related concepts is responsible for this. Even if he should have used AI to formulate his question, the result is not an really incoherent "AI slop". Is it TLDR? Not as a single question, only if he would regularly post such long questions, or long answers.

 

[gentzen]

(1) Decoherence and symmetry:
Am I right in understanding that the asymmetry of the environment in standard decoherence models reflects a common modeling choice (e.g., heat baths, laboratories, etc.), rather than a fundamental requirement of quantum mechanics itself? In other words, is there anything in standard QM or relativistic QFT that forbids a globally closed system from remaining exactly symmetric at the level of the total quantum state, even if reduced subsystems exhibit effectively classical behavior?

As long as you are requiring exact symmetry, my answer to QuarkyMeson above provides one answer: If everything is exactly symmetric down to the last detail, then one can simply take the quotient and get a reduced description.
So this means that only the FAPP situation is interesting for your question. It can still be an exact symmetry nearly everywhere, but at least the state of the environment should not be exactly symmetric, but only symmetric FAPP. And then one can try to study whether symmetry breaking FAPP occurs in that scenario.

 

[jj90]

Thanks everyone - this is exactly the clarification I was looking for.

Regarding the “AI slop” comment: I used AI only to help reword and tighten the question for clarity and concision, not to generate the underlying content or reasoning. As Gentzen noted, my aim was simply to understand whether the symmetry assumptions I described are coherent in principle within standard QM/QFT.

I appreciate everyone taking the time to engage with the question!

 

[QuarkyMeson]

So the small part broke the symmetry, no? And not just for itself, but globally! Or maybe not, if we assume that another indistinguishable small part also exist, as required by the symmetry under that 180° spatial rotation about a fixed axis.
The interesting question is what "indistinguishable" means or implies in this context. In the end, it probably means that we are allowed to take the quotient by the symmetry, and end up in a situation where only one small part is left, and the description is reduced.

Did it? I was under the assumption that parts of the system interacting with each other doesn't break the symmetry of the whole. If the rules respect the symmetry, the overall state keeps respecting it too, no matter how messy the internal dynamics get. Entanglement and correlations can make subsystems look asymmetric after you ignore degrees of freedom or condition on an outcome, but that’s not the same thing as the global symmetry being violated.

So, if closed evolution is unitary, symmetries that commute with the Hamiltonian are preserved under unitary evolution, everything that looks like symmetry breaking, decoherence, irreversibility, etc. is what you get when you restrict your description to a subsystem by tracing out the rest. Aka, to fix it, just go to the church of the larger Hilbert space. This was partly my understanding of some of the motivation for quantum channels (CPTP maps). Wouldn't be the first time I was mistaken.

I have the impression that jj90 genuinely wants to understand whether the symmetry involving his "two arbitrarily complex macroscopic subsystems" would break, and whether one the given six quantum-related concepts is responsible for this. Even if he should have used AI to formulate his question, the result is not an really incoherent "AI slop". Is it TLDR? Not as a single question, only if he would regularly post such long questions, or long answers.

I had a different impression, but I'll be the first admit that I might have judged too harshly.