# Do objective collapse equations actually collapse the state?

Why are objective collapse theories stated to collapse the state from a superposition to a single eigenstate (corresponding to the measured eigenvalue)? For this discussion, we are focusing on the mathematics of the equations, not on any interpretation on quantum mechanics. I merely wish to know if objective collapse equations (the dynamics as described by the typically stochastic differential equations) yield something like: (a) $$$$\langle x\mid \rho(t)\mid y\rangle \propto e^{-\lambda(x-y)^2 t}\langle x\mid \rho(0)\mid y\rangle$$$$ or (b) $$$$\langle x\mid \rho(t)\mid y\rangle \propto e^{-\lambda((x-x_0)^2+(y-x_0)^2) t}\langle x\mid \rho(0)\mid y\rangle$$$$ where the system tends to the pure state $$\mid x_0\rangle \langle x_0\mid$$, where $$x_0$$ is the randomly measured eigenvalue.

In (a), we have a process which can be achieved via a Lindblad equation derived from an open quantum system. In this context the process would be termed "decoherence".If we find (a), then I find the name "objective collapse" to be inappropriate because I would define collapse as in (b). Hence, the second question is whether anyone actually uses the term "collapse" for the process in (a).

• these theories are not mainstream physics . "According to collapse theories, energy is not conserved, also for isolated particles. " en.wikipedia.org/wiki/… Sep 22, 2023 at 4:20
• Dynamical collapse theories don't eliminate superpositions, they just reduce the amplitude of other states. This is different from decoherence, which doesn't necessarily reduce the amplitude of other states in a superposition. The effects of collapse may have detectable effects in the low amplitude states. See arxiv.org/abs/1407.4746 Sep 22, 2023 at 6:52
• I disagree with the idea that objective collapse theories are not mainstream physics. We have a lot of questions about Bohmian mechanics, for example, which is not any more 'mainstream'. I vote to reopen. Sep 22, 2023 at 8:50
• Objective collapse theories are discussed in many mainstream journals. For example, there are hundreds of papers on Continuous Spontaneous Localization in journals such as Physical Review A & D, Physics Letters A, Nature Scientific Reports, …. Sep 22, 2023 at 12:10
• @DavidBailey i think the reason they are not considered mainstream is because they make predictions that differ from standard quantum mechanics which is why I have seen them called extensions of qm rather than interpretations some people even say that they can only be true if quantum mechanics itself is false. preposterousuniverse.com/blog/2013/01/17/… arxiv.org/abs/2307.05153 quantamagazine.org/… May 10 at 2:14

There are three ways to read this question, depending on your perspective about wavefunctions and density matrices. Here are three perspectives you might take, listed in order from most common to least common.

1. There is in fact one objective wavefunction, from which you can construct one objective pure-state density matrix. A mixed-state density matrix is not an objective state of reality, but rather a (subjective) state of knowledge or belief, indicating how one would assign probabilities given limited knowledge about the actual full wavefunction.

2. The wavefunction itself is subjective, and so is the associated pure-state density matrix. A mixed-state density matrix is evidently at least as subjective as a pure-state, if not more so.

3. There is one objectively correct density matrix in any situation, even if it is in a mixed state.

I'm not sure your question makes much sense from perspective 1). You are not really distinguishing between wavefunctions and density matrices, and you are talking about an "objective" collapse as ending up with a mixed state density matrix, so I suspect you're not thinking of this density matrix as being subjective. (?)

Perspective 2) also seems like there would be a disconnect. If everything in ordinary quantum theory is subjective, then evidently there would be a world of difference between an "objective collapse" and mere subjective uncertainty.

Perspective 3) is quite uncommon, but I suspect that's the best way to make sense of your question. This perspective grapples with the idea that even a mixed state could somehow be an objective state of affairs. I've never been able to wrap my head around this perspective very well. (What does probability even mean in such a situation? What does objectivity mean?) Still, I guess I can almost see how (given this viewpoint) an objective collapse could be said to land you into an "objectively mixed state".

But even if your perspective is 3), then there is still a huge difference between this and decoherence. For this discussion, please see the answers to this previous question. These answers might also help if you are taking perspective 1).

• Yes, (3) would be my perspective, since I am interested in a pure mathematical perspective of what the dynamical equation yield. I am also intrigued by implication that "thinking of a mixed state as being something physical" is seldomly done. When we are talking about any subsystem of a pure state total system, even excluding dynamics, we are typically led to a mixed state. As such, the mixedness is something real when focusing on a subsystem. Why would focusing on a subsystem automatically mean that one now has to only consider the pure states? Sep 22, 2023 at 15:43
• Let's say there are two observers; one chooses a subsystem A, and another chooses a subsystem B which mostly overlaps with A, but not quite. Each of them now has a different subjective density matrix that mostly refers to the same stuff. How could each of them consistently implement an 'objective wavefunction collapse" on their own density matrix, in a way that agreed with the other? Seems pretty subjective at that point. To say that a matrix is "something real" when focusing on a chosen subsystem seems to me to be a mis-use of the word "real". Sep 23, 2023 at 14:03