I wonder if it makes sense to say two physical systems are isomorphic to each other. Say if I have a system of electron spins in a magnetic field, and another system with an ammonia molecule. Since both systems have two basis quantum states (or by approximation), can I say the two systems are isomorphic to each other, or there's an isomorphism of their eigenstates?


1 Answer 1


Based on the example in the question, I'm interpreting physical system to mean mathematical model of a physical system (also called a theory). That's important, because isomorphism is a mathematical concept — but I'll bend that rule a bit at the end of this answer.

An isomorphism is an invertible map between two things that preserves the structure(s) of interest. The same two things can be both isomorphic and non-isomorphic to each other, depending on which structures we want to preserve.

If the Hilbert space is the only structure of interest, then for any given $N$, all system with $N$-dimensional Hilbert spaces are isomorphic to each other.

For physics, we also care which operators represent observables, and we care about time. At a minimum, for each time-interval $T$, a model should specify which operators $\Omega(T)$ represent observables at those times. (I'm using the Heisenberg picture, which is usually the best picture for conceptual clarity even if it's not always the best picture for calculation.) A more refined concept of isomorphism is one that preserves this structure. Explicitly: An isomorphism between models $1$ and $2$ is a unitary transformation $U$ from one Hilbert space to the other, such that $U\Omega_1(T)=\Omega_2(T)U$ for all time-intervals $T$.

In quantum field theory (QFT), we care about the association between observables and regions of spacetime, not just time-intervals. An even more refined concept of isomorphism would thus be to require $U\Omega_1(R)=\Omega_2(R)U$ for all spacetime regions $R$. Assuming that spacetime has some geometric symmetries, like Poincaré symmetry, we could improve this concept of isomorphism by allowing $U\Omega_1(R)=\Omega_2(R')U$ where $R\to R'$ is an isometry of spacetime.

In mathematical physics, especially in the study of 't Hooft anomalies, a different definition of QFT is often used. Roughly, a QFT is a functor from a category of structure-preserving bordisms, such as spacetime-metric-preserving bordisms, to a category of Hilbert spaces. (I listed some references in another question.) In this formulation, we could call two models isomorphic if the functors are naturally isomorphic, using category theory's off-the-shelf definition of natural isomorphism.

If a theory is meant to be applied to the real world, then its postulates must specify some kind of association between elements of the mathematical structure and elements of the real world. In quantum theory, this means specifying which operators represent which real-world observables (measurable things). We might try to use a definition of isomorphism that enforces that association, but that's going beyond the scope of pure math, so making it precise is probably difficult. On the other hand, such a concept of isomorphism probably isn't needed, because two models that are isomorphic in that sense are really the same model in absolutely every way that matters for physics, so we might as well just call them same instead of isomorphic.

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    $\begingroup$ Thanks so much for the great explanation:) $\endgroup$
    – IGY
    Jul 18, 2021 at 16:40
  • $\begingroup$ I have a question about the distinction between equivalence of theories vs. "objects"? For some purposes, all gases of particles with 5 degrees of freedom can be put into one equivalence class, yet you wouldn't say a gas of oxygen molecules is the same as a gas of nitrogen molecules - because there exist other theories in which they would not be put into one equivalence class. When you say there is no point in specifying what real-world systems the "isomorphic" theories are associated with, is that meant for the fundamental theory rather than effective theories, e.g. statistical physics? $\endgroup$
    – Lili FN
    Jul 18, 2021 at 20:32
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    $\begingroup$ @LiliFN That's not what I meant to say. If two mathematically-identical theories are associated with different real-world systems, then they are different theories. They are isomorphic mathematically, but different physically. If two models have the same mathematical structure and also have the same physical interpretation, then we might as well just call them the same, without any qualification. That's what I meant to say. The word isomorphic is normally used when they are the same in some ways but might still be different in other ways. $\endgroup$ Jul 18, 2021 at 20:50
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    $\begingroup$ @Chiral Anomaly thanks for clarifying! $\endgroup$
    – Lili FN
    Jul 18, 2021 at 20:51

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