Gibb's Paradox, Interaction States and Quantum Mechanics?
Francesco R. Ruggeri
Gibb’s Paradox is associated with introducing a 1/N! factor in the partition function= 1/N! {Sum over i exp(-ei/T)} power N (1). It is often stated that N! is required to deal with identical particles. It is further stated that in classical physics, particles have an identity, but in quantum mechanics, they are identical and so the N! is really a quantum mechanical feature. In (2), we argued that dE=TdS - PdV is entirely focused on interactions and so if S = -k ln(number of states), one should consider interaction states, not identity states. We argued that a green particle differs in colour from a red one, but this has nothing to do with interactions and one would completely ignore colour when calculating the number of states. The same reasoning applies to two identifiably different particles with the same e,p being at x1, and x2 versus x2, x1. In terms of identity states, there are two, but they both represent the same physical interaction and are counted once. Even though classical particles have an identity, and quantum ones also because they have a center-of-mass, from the point of view of an interaction, a particle at (x,y,z) with a certain momentum and energy is all that one needs to know for interaction purposes. It is irrelevant which particle is at this (x,y,z) and so N! Is introduced for this reason, as argued in (2). Here we consider the quantum mechanical case. We have argued in previous notes that quantum mechanics follows from special relativity. In particular, a particle still has a center-of-mass and so an identity. If it moves with constant speed, one has the classical equation x=vt for the center-of-mass. This begs the question: Where are the quantum mechanical features? We argue they are in the interaction. The fact that the center-of-mass is at some x,t does not mean that a p (momentum) impulse hit is rendered at that point. In fact, the Lorentz invariant -Et+px, which is more general than x=vt, may be considered to be a kind of free particle potential with x,t and x+hbar/p, t+hbar/E having the same potential values and rendering the same hit, regardless of whether x=vt. -Et+px is already created using the notion x=vt for the center-of-mass and so -Et+px holds for more x, t points than simply x=vt. As a result, the quantum free particle wavefunction exp(-iEt+ipx) represents a probability to interact, not a center-of-mass identity of the particle. In other words, quantum mechanics is a statistical theory of interaction and as such the probability bypasses the actual position of the center-of-mass which identifies the particle. In a 2-slit experiment, a particle has a specific center-of-mass and moves as x=vt, but at the 2-slit apparatus (with slits about hbar/p apart), it may probabilistically interact with both slits. This is a probabilistic interaction calculation which has little to do with the center-of-mass position and hence any position identity of the particle. As a result, a more general quantum wavefunction W(x)= Sum over p a(p)exp(ipx) (bound state) is also an interaction probability. Thus, quantum mechanics automatically only considers interactions and this leads some to argue that there is no particle identity in quantum mechanics. We argue that just like classical physics, there is a particle identity (center-of-mass), but the interaction is the main focus and does not depend on this identity position. In other words, one only deals with interaction probabilities, just like classical statistical mechanical problems should only deal with interaction states (as argued in (2)), and not with any physical state which may be identified, but does not change the interaction. The difference between quantum mechanics and classical mechanics, is that quantum mechanics is formulated on the notion of interaction (so the particle identity is removed from the calculation right away), while in the classical case one retains the identity and has to manually correct for identity states in order to convert the number to the number of interaction states.
•Mar 5, 2026•0 citations