The apparent clash between Leibniz Principle of Identity of Indiscernibles PII and the Copenhagen Interpretation of Quantum Mechanics (StdQM) has triggered quite a bit of discussion surveyed in the book Identity in Physics: A Historical, Philosophical and Formal Analysis.
The trouble is rooted in the interpretation of the wave function of stdQM as expressing probabilities of possible electron particle configurations.
This is to be compared with actual real configurations as in Real Quantum Mechanics RealQM in a sense of classical physics with non-overlapping charge densities with unique presence in space-time as expression of identity.
PII is in harmony with classical physics and RealQM, but not with StdQM.
Schrödinger as inventor of quantum mechanics could not accept the probabilistic interpretation of StdQM, and so was cancelled by the leading Copenhagen school of Bohr, Born and Heisenberg.
We may ask if PII is of real importance or only of some scholastic philosophical virtual importance?
The previous post brought up the idea that PII connects to self-interaction as a toxic element of Quantum Field Theory QFT as the generalisation of StdQM underlying the Standard Model capturing all of elementary atomic particle physics. It is manifested in the appearance of "infinities" asking for "renormalisation" to be cancelled, like techniques to ignore elephants in the room.
In classical physics prevention of self-interaction is possible because it is possible to distinguish each particle from all other particles and so to guarantee in particular that the electric/gravitational field created by a particle only affects other particles but not itself. This is the nature of Newton's Law of gravitation and Coulomb's Law.
But StdQM describes probabilities of possible particle configurations, which lack particle paths and so lack identity over time. In StdQM bosons (such as photons) can occupy the same position in space-time as well as some fermions (such as electrons with different spin), and particle paths have no meaning. In this setting self-interaction cannot easily be prevented, and so ask for extra-ordinary techniques for cancellation in the form of "renormalisation". Nobody is happy with this trick introduced to handle a fundamental difficulty of physics as statistics.
The possibility that a specific particle occupies some specific position in space-time and the possibility that another particle does the same thing do not appear to be mutually exclusive, which means that particle identity is lost. Probably. Statistics is tricky.
The problem with self-interaction is that it has to steer way from both blow-up to infinity (too much ego) or decay to zero (too much self-criticism) in a very delicate balance threatened by instability.
Recall that the electron of Hydrogen atom is prevented from disappearing into the potential hole of the proton kernel by the presence of the Laplacian in Schrödinger's equation giving the electron an extension in space as a charge density. Likewise the Earth is saved from being swallowed by the Sun by orbiting the Sun as a form of spatial extension.
From the above book:
- It is not clear how collections of non-individual objects can be captured by standard set theory.
- As the mathematician Yuri Manin put it: “We should consider possibilities of developing a totally new language ...” to deal with collections of entities which do not behave as standard sets (in the sense of obeying the axioms of the usual set theories), since the “new quantum physics has shown us models of entities with quite different behaviour.
- Even ‘sets’ of photons in a looking-glass box, or of electrons in a nickel piece, are much less Cantorian than the ‘set’ of grains of sand”.
- It is our intention in this book to explore these different issues and, in particular, to go some way towards developing the ‘totally new language’ suggested by Manin.
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