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.