Representation Theory

This is a deep and well-posed question. Let me try to lay out the abstract machinery first, then bring in the observer, then address the dimensionality mismatch you're pointing at.

1. The abstract root: groups, actions, representations

At the root, representation theory is the study of how a group $G$ acts on a vector space $V$ by linear maps:

$$\rho :G\to GL(V),\rho (g_1{g}_2)=\rho (g_1)\rho (g_2)\mathrm{.}$$

The group is the abstract structure — the symmetries. The representation is a concrete realization of that structure as transformations of a space. The dimension of $V$ is the dimension of the representation.

The key theorem (for compact groups, and for finite groups) is that every representation decomposes into irreducible representations (irreps), and the irreps are the atoms. Everything else is a direct sum of irreps. So the "parts" that get reused across physics are the irreps.

2. Why irreps are the natural atoms

An irrep is a representation with no proper invariant subspace. It cannot be broken down further. Schur's lemma says that the only operators commuting with an irrep are scalars. That rigidity is why irreps are stable across contexts: once $SO(3)$ acts on a space, the space must decompose into irreps of dimensions $1,3,5,7,\dots$, regardless of what physical system you're describing. The physics chooses which irreps appear; the mathematics fixes what irreps are available.

3. The observer as a representation

Here is where your question becomes sharp. If the observer is also a physical system, then the observer is also described by a representation of the same symmetry group. The observer has:

• an internal state space $V_O$,

• a group action ${\rho }_O:G\to GL(V_O)$,

• a decomposition into irreps.

The observed system has:

• a state space $V_S$,

• a group action ${\rho }_S:G\to GL(V_S)$,

• its own decomposition.

The joint system is $V_O\otimes V_S$, and the group acts by ${\rho }_O\otimes {\rho }_S$. The joint representation decomposes by the Clebsch–Gordan rules:

$$V_O\otimes V_S\cong \underset{i}{⨁}{V}_i\mathrm{.}$$

This is the abstract heart of what you're asking. The observer and the observed are not separate representations that happen to interact. They are factors of a tensor product, and the tensor product decomposes into irreps according to fixed rules.

4. The dimensionality mismatch

You're pointing at something real: the observer's representation space and the observed system's representation space generally have different dimensions. The observer might be a spin-$1\mathrm{/}2$ system (dimension 2), the observed system might be a spin-$1$ system (dimension 3). The joint space has dimension $2\times 3=6$, which decomposes as

$$2\otimes 3=4\oplus 2$$

in $SU(2)$ language — a spin-$3\mathrm{/}2$ and a spin-$1\mathrm{/}2$.

So the dimensionality of the object relative to the observer is not a fixed ratio. It is mediated by the tensor product and its decomposition. The observer does not need to have the same dimension as the observed. What matters is that both carry representations of the same group, and the joint representation is well-defined.

5. Where the mismatch becomes physical

There are several places where this dimensionality question has teeth:

Measurement. In quantum measurement, the observer's state space must be large enough to encode the outcome. A two-dimensional observer (qubit) can distinguish two outcomes. To distinguish $n$ outcomes, the observer needs at least $n$ orthogonal states. So the dimensionality of the observer bounds the resolution of the observation.

Encoding. The observer's representation of the world is a subspace of the joint space. The map from system to observer is a channel, and its capacity is bounded by the dimension of the observer's space. This is the beginning of quantum information theory.

Symmetry breaking. If the observer has a preferred orientation (a pointer state), that breaks the symmetry. The observer's representation is then not the full irrep but a subrepresentation or a coherent state — a point in the orbit of the group action. The observed system, interacting with this oriented observer, sees a broken-symmetry environment.

6. The categorical view

At the most abstract level, both observer and observed are objects in a category of representations, and the interactions are morphisms (intertwiners). The tensor product is a monoidal structure. The irreps are simple objects. The Clebsch–Gordan decomposition is the monoidal product's structure.

In this language, your question becomes: given two objects $V_O$ and $V_S$ in $\mathrm{R}\mathrm{e}\mathrm{p}(G)$, what is the structure of $V_O\otimes V_S$, and how does the observer's dimension constrain the possible morphisms $V_S\to V_O$?

The answer is: the possible intertwiners are exactly the Clebsch–Gordan coefficients, and they exist only between matching irreps. The observer can only "see" the components of the system that have overlap with its own representation.

7. The dimensionality of the object relative to the observer

This is the sharpest version of your question. Let me state it precisely:

If the observer's state space has dimension $d_O$ and the system's has dimension $d_S$, what is the dimension of the system as represented in the observer?

The answer depends on the map. If the observer is a perfect encoder, the represented dimension is $\min (d_O,d_S)$. If the observer is a projection, it is the rank of the projection. In general, the represented dimension is the rank of the intertwiner between the system's representation and the observer's representation.

So the object does not have a single dimensionality relative to the observer. It has a spectrum of dimensionalities, one for each irrep in the decomposition, filtered by the observer's capacity to represent that irrep.

8. The deep point

The deep point is that dimensionality is not absolute. It is relative to a representation, and a representation is relative to a group, and a group is relative to a symmetry. The observer and the observed are both representations, and their relationship is mediated by the tensor product and its decomposition into irreps.

So when you ask "what is the dimensionality of the object with regard to the dimensionality of the observer," the answer is: it is the dimension of the intertwiner space between their representations. That is the mathematically precise version of your intuition that they define each other.

9. Where this connects to physics

• In quantum reference frames, the observer's orientation is a physical degree of freedom, and the observed system's state is defined relative to it. The dimensionality of the reference frame bounds the distinguishability of the system's states.

• In relational quantum mechanics, states are always relative to an observer, and the observer is always a physical system with its own representation.

• In holography, the boundary theory has fewer dimensions than the bulk, and the map between them is a representation-theoretic encoding. The dimensionality mismatch is the whole content of the duality.

If you want, I can go into the Clebsch–Gordan decomposition in detail, or into how quantum reference frames formalize the observer as a representation, or into the categorical formulation of observer–observed relations.