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.