- #1
Iamu
- 24
- 2
I've been thinking about the many-worlds interpretation and how one might test it experimentally. I'm wondering if it might be possible to observe interference between macroscopic systems in different "worlds".
We start with an isolated quantum system in superposition, and we let it interact with a macroscopic measuring device. If I understand correctly, according to the Copenhagen interpretation, we have collapsed the wavefunction of the system and it now exists in a single well-defined state, while according to MWI, we have entangled the system with the measuring device and together they exist in a superposition of states. So it seems that it may be possible to observe some additional interference in the MWI case except that this is hindered by decoherence, while according to the Copenhagen interpretation, once the measurement is made, things simply are what they are and there is no further opportunity for interference to be observed.
In MWI, the states most probably quickly decohere and no interference between the alternatives states of the measuring device can be observed; that is to say, the hidden degrees of freedom in the measuring device evolve randomly in each case of an alternative measurement and reduce the chances for interference between the superposed states to approximately zero.
But a logical rammification of MWI, is that for each possible evolution of the hidden degrees of freedom in the measuring device after the measurement is made, there should be a small subset of worlds in which every hidden degree of freedom has evolved in the exact same manner as in other worlds where the measurement returned a different result. This may be a technically impossible idealization, because the act of measuring may influence many hidden degrees of freedom, but I don't think this necessarily matters. Even if there is a difference in these hidden degrees of freedom, if the difference is small compared to the total degrees of freedom in the system, then the difference in the scalar products between the state vectors could be made arbitrarily small. Then, interference effects between these particular subsets of worlds could be observed. In other words, if [tex]|\psi>[/tex] is the state of the system being observed, and [tex]|\epsilon>[/tex] is the state of the environment or the measuring device, then the probability of transition from [tex]\psi[/tex] to [tex]\phi = |<\phi|\psi>|[/tex]2 = [tex]|\Sigma[/tex]i[tex]\psi[/tex]i*[tex]\phi[/tex]i|2 before the system and device are entangled and [tex]\Sigma[/tex]j[tex]|<after|\phi,\epsilon[/tex]j>|2 = [tex]\Sigma[/tex]j[tex]|\Sigma[/tex]i[tex]\psi[/tex]i[tex]*<i|\phi><\epsilon[/tex]i[tex]|\epsilon[/tex]j[tex]>|[/tex]2 afterwards. Generally, [tex]<\epsilon[/tex]i[tex]|\epsilon[/tex]j[tex]>[/tex] is taken to be approximately the Kronecker delta and then cross-terms cancels and probabilities become additive, but I think it can be made arbitrarily close to one by increasing the hidden degrees of freedom and/or reducing the effect of measurement on the system and only considering worlds which are "close enough" to the one in question, and then the last expression reduces to the first and interference terms are recovered.
I'm only an undergrad physics student and my knowledge of QM and math is limited, so I'll defer to other posters on this one; did I mangle this, or is it reasonable to think that we could replace the Kronecker delta with 1 by only considering "close enough" worlds? Do we have to consider all parts of the wavefunction, or can we select just the "close enough" parts that interest us in this case?
I'll withhold my speculation on how we could design an experiment to test MWI until after I hear other's opinions on this.
We start with an isolated quantum system in superposition, and we let it interact with a macroscopic measuring device. If I understand correctly, according to the Copenhagen interpretation, we have collapsed the wavefunction of the system and it now exists in a single well-defined state, while according to MWI, we have entangled the system with the measuring device and together they exist in a superposition of states. So it seems that it may be possible to observe some additional interference in the MWI case except that this is hindered by decoherence, while according to the Copenhagen interpretation, once the measurement is made, things simply are what they are and there is no further opportunity for interference to be observed.
In MWI, the states most probably quickly decohere and no interference between the alternatives states of the measuring device can be observed; that is to say, the hidden degrees of freedom in the measuring device evolve randomly in each case of an alternative measurement and reduce the chances for interference between the superposed states to approximately zero.
But a logical rammification of MWI, is that for each possible evolution of the hidden degrees of freedom in the measuring device after the measurement is made, there should be a small subset of worlds in which every hidden degree of freedom has evolved in the exact same manner as in other worlds where the measurement returned a different result. This may be a technically impossible idealization, because the act of measuring may influence many hidden degrees of freedom, but I don't think this necessarily matters. Even if there is a difference in these hidden degrees of freedom, if the difference is small compared to the total degrees of freedom in the system, then the difference in the scalar products between the state vectors could be made arbitrarily small. Then, interference effects between these particular subsets of worlds could be observed. In other words, if [tex]|\psi>[/tex] is the state of the system being observed, and [tex]|\epsilon>[/tex] is the state of the environment or the measuring device, then the probability of transition from [tex]\psi[/tex] to [tex]\phi = |<\phi|\psi>|[/tex]2 = [tex]|\Sigma[/tex]i[tex]\psi[/tex]i*[tex]\phi[/tex]i|2 before the system and device are entangled and [tex]\Sigma[/tex]j[tex]|<after|\phi,\epsilon[/tex]j>|2 = [tex]\Sigma[/tex]j[tex]|\Sigma[/tex]i[tex]\psi[/tex]i[tex]*<i|\phi><\epsilon[/tex]i[tex]|\epsilon[/tex]j[tex]>|[/tex]2 afterwards. Generally, [tex]<\epsilon[/tex]i[tex]|\epsilon[/tex]j[tex]>[/tex] is taken to be approximately the Kronecker delta and then cross-terms cancels and probabilities become additive, but I think it can be made arbitrarily close to one by increasing the hidden degrees of freedom and/or reducing the effect of measurement on the system and only considering worlds which are "close enough" to the one in question, and then the last expression reduces to the first and interference terms are recovered.
I'm only an undergrad physics student and my knowledge of QM and math is limited, so I'll defer to other posters on this one; did I mangle this, or is it reasonable to think that we could replace the Kronecker delta with 1 by only considering "close enough" worlds? Do we have to consider all parts of the wavefunction, or can we select just the "close enough" parts that interest us in this case?
I'll withhold my speculation on how we could design an experiment to test MWI until after I hear other's opinions on this.