No geometry or topology can save the collapse of the wave function

 Short answer: yes — but each time we try, the “collapse problem” does not simply vanish; instead it turns into a different problem.

There are spaces and topologies where collapse becomes natural, emergent, or unnecessary, but in every case the issue reappears in a new form. The measurement problem is extremely stubborn.

Let me organize the possibilities clearly.

✅ 1. Projective Hilbert Space: Collapse becomes a geometric “jump,” but the problem remains

Projective Hilbert space
\[
\mathbb{P}(\mathcal{H})
\]
is the true space of physical quantum states.
It is a smooth manifold equipped with the Fubini–Study metric.

In this space, collapse looks like:

A discontinuous jump from one point on the manifold to another.

This makes collapse cleaner geometrically, but:

• It still violates smooth Schrödinger evolution.
  
• It is still instantaneous and global.
  
• It still needs a trigger (measurement).
  

The topology didn’t fix the collapse; it just rephrased it.

✅ 2. Rigged Hilbert Spaces: collapse becomes mathematically cleaner, but not conceptually solved

A rigged Hilbert space (Gelfand triple):

\[
\Phi \subset H \subset \Phi^\*
\]

adds distributions, Dirac deltas, momentum eigenstates, etc.

This lets “collapsed” states (like exact eigenvectors) be well-defined objects.

This removes technical problems:

• no worries about non-normalizable eigenstates,
  
• collapse maps states into elements of a larger dual space.
  

But it does not fix:

• the non-unitary jump,
  
• the physical meaning of measurement,
  
• the instantaneous global change.
  

It solves functional analysis issues, not the measurement problem itself.

✅ 3. Decoherence Space (pointer basis formulation): collapse becomes effectively unnecessary

If you enlarge the space to include:

\[
\mathcal{H}{\text{system}} \otimes \mathcal{H}{\text{environment}},
\]

then tracing over the environment produces effective collapse without requiring an actual physical jump.

In this larger tensor-product Hilbert space:

• Schrödinger evolution is always unitary.
  
• Apparent collapse in the system’s reduced density matrix arises from decoherence.
  

But:

• there’s still no explanation of why only one outcome is observed
  (the “problem of outcomes”).
  
• decoherence gives apparent collapse, not real collapse.
  

So the problem is softened, not eliminated.

✅ 4. Many-Worlds (Everett): the topology eliminates collapse entirely

Many-Worlds says:

• the universal state lives in Hilbert space,
  
• the state never collapses,
  
• the wavefunction always evolves unitarily.
  

Here the space is simply:

\[
\mathcal{H}_{\text{universe}}
\]

and collapse does not exist as a process.

But the problem becomes:

• What is a branch?
  
• How do probabilities arise from a deterministic wavefunction?
  
• Why does an observer experience one branch?
  

So collapse is not a problem anymore — but something else is.

✅ 5. Objective Collapse Models (GRW / CSL): collapse becomes a dynamical process

These approaches add a new topology on state space — stochastic discontinuous trajectories.

In GRW the state lives in the same Hilbert space, but evolves via:

\[
d|\psi\rangle = \text{(Schrödinger evolution)} + \text{(random jumps)}.
\]

Here:

• collapse is built into the space’s dynamics, not an ad hoc rule.
  
• collapse is no longer a problem — it’s just how the space behaves.
  

But this introduces:

• new constants,
  
• new physical processes,
  
• energy non-conservation issues,
  
• and the need for experimental tests.
  

The collapse problem is replaced by the “what new physics have we introduced?” problem.

✅ 6. Algebraic Quantum Theory: states live on C*-algebras, not Hilbert space

In algebraic quantum mechanics:

• the primary object is a C*-algebra of observables,
  
• states are positive linear functionals,
  
• Hilbert spaces arise only after choosing a representation.
  

Collapse becomes:

Bayesian updating of a state functional after acquiring new information.

Here the topology is more general (weak-* topology), and collapse looks more natural.

However:

• one still has to explain why a definite outcome occurs,
  
• updating still produces a discontinuous change in the state,
  
• “measurement” still requires interpretation.
  

⭐ **7. The one general lesson:

Changing the space changes the form of the collapse problem, not its existence**

There is no known space or topology that:

• preserves the empirical content of quantum mechanics,
  
• retains the probabilistic predictions,
  
• and eliminates all conceptual problems related to collapse.
  

But different spaces do change which part of the problem becomes easier:

Space / Topology	What becomes easy	What remains hard
Projective Hilbert	Geometry of states	Discontinuity, outcomes
Rigged Hilbert	Eigenstates as real objects	Measurement meaning
Decoherence (tensor-product)	Effective collapse	Why one outcome?
Many-Worlds	No collapse	Probability interpretation
GRW/CSL	Physical collapse model	New physics required
C*-algebraic	Bayesian update	Definite outcomes

🎯 Final answer:

We can choose spaces and topologies where collapse becomes:

• unnecessary (Many‑Worlds),
  
• emergent (Decoherence),
  
• dynamical (GRW),
  
• or mathematically routine (Rigged Hilbert space).
  

But there is no known space where the entire collapse problem simply “goes away.”
It always reappears as a different conceptual issue.

If you want, we can go deeper into:

• which topology gives the cleanest version of measurement,
  
• whether quantum state space can be embedded in a classical phase space,
  
• whether collapse naturally disappears in geometric quantum mechanics,
  
• or whether a “bigger” space (like category-theoretic or topos-theoretic) can resolve collapse.
  

Just tell me which direction you want!