Abstract
Chapter 3 focuses on the problem of finding a justification of the Symmetrization Postulate over and above the typical arguments based on the Indistinguishability Postulate. One particular argument, known as the exchange degeneracy argument, is carefully evaluated. It is claimed that this argument must be based on an interpretational rule which we dub “Ontic (or Epistemic) Conservativeness of Superpositions”. The notion of permutation is also discussed at length, with an emphasis on the philosophical issue of modality de re. In addition, the chapter contains a formal and conceptual discussion of the notion of paraparticles, that is, purely hypothetical particles whose states display types of symmetries other than bosonic and fermionic. We present some theoretical reasons why paraparticles may not be observable in nature.
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Notes
- 1.
For a comprehensive formal and philosophical analysis of the notion of permutation symmetry in QM, see French and Rickles (2003).
- 2.
If among the physical processes we include measurements under the collapse interpretation, then it is straightforward to give a physical process that can transform a product state into a symmetric/antisymmetric one. Suppose we start with a product state |φ〉|ψ〉 where |φ〉 and |ψ〉 are orthogonal. Let us define the following vectors: |χ〉 = \( \frac{1}{\sqrt{2}}\left(|\left.\varphi \right\rangle |\left.\psi \right\rangle +|\left.\psi \right\rangle |\left.\varphi \right\rangle \right) \) and |η〉 = \( \frac{1}{\sqrt{2}}\left(|\left.\varphi \right\rangle |\left.\psi \right\rangle -|\left.\psi \right\rangle |\left.\varphi \right\rangle \right) \). Then it can be easily verified that the symmetric projectors Eχ ⊗ Eη + Eη ⊗ Eχ and Eχ ⊗ Eχ (where Eχ = |χ〉〈χ| and Eη = |η〉〈η|) project the original state |φ〉|ψ〉 onto, respectively, the antisymmetric and symmetric sectors of the total Hilbert space. And since these projectors represent possible outcomes of measurements for admissible observables, it is physically possible to put the system into an antisymmetric/symmetric state by means of measurement procedures.
- 3.
- 4.
The same applies in the case when the particles are distinguishable, but their identifying properties are dynamically irrelevant. See Saunders (2015, p. 176ff) for a discussion of the classical case of permutation invariance.
- 5.
This contradiction can be avoided by limiting admissible observables to the symmetric ones, though, since in that case there will be no experimental procedure that could differentiate between permuted states. Thus the conditional statement regarding the zero probability of finding the system in state |ψ〉1|φ〉2 given that it occupies state |φ〉1|ψ〉2 would be vacuously true, since the antecedent (“If we performed such-and-such measurement, then…”) would be necessarily false. Still, as a matter of formal elegance, it is better not to have both permuted states |φ〉1|ψ〉2 and |ψ〉1|φ〉2 in one representational framework.
- 6.
This probability can be equivalently presented as the expectation value for the projector onto the ray spanned by the ket \( \frac{1}{\sqrt{2}}\left(\left|\varphi \Big\rangle +|\psi \right\rangle \right)\otimes \frac{1}{\sqrt{2}}\left(\left|\varphi \Big\rangle +|\psi \right\rangle \right) \).
- 7.
The main obstacle on a path leading from the Born rule to OCS is the notorious gap between probability one and pre-measurement reality. The Born rule guarantees that it is certain that upon measurement the system will find itself in a particular physical state described as either |φ〉 or |ψ〉, but from this it does not logically follow that the system already occupies this state before measurement. We have to be wary of this type of reasoning, lest we are forced to accept the logical transition from the fact that it is guaranteed (with probability one) that a certain definite value of an observable will be revealed to the conclusion that a certain definite value of this observable is possessed before measurement.
- 8.
This method generalizes to the case of N particles as follows: for the subspace of ℋN, we select the basis vectors from the set {|i1, i2, …, iN〉: i1 ≤ i2 ≤ … ≤ N and ik = 1, 2, …, n}. Thus we take all non-descending N-element sequences of numbers from 1 to n to form the required basis.
- 9.
Formally, operations Sym and Anti can be treated as projection operators, projecting onto symmetric and antisymmetric subspaces of ℋ 3.
- 10.
This is just a coincidence caused by the fact that the number of dimensions of ℋ and the number of factors in the product of Hilbert spaces happen to be identical. In the general case when the number of particles is N and the dimensionality of each Hilbert space is K (K > N), there will be \( \left(\begin{array}{c}K\\ {}N\end{array}\right) \) orthogonal antisymmetric vectors (the number of N-element combinations out of K distinct elements).
- 11.
The approach used below is closely modeled on Peres (2002, p. 131ff).
- 12.
To see that, take the inner product of (3.3) and Sym|αβγ〉 (Anti|αβγ〉).
- 13.
The fact that the nth roots of unity form a cyclic group Zn may be used as an explanation of why there are no one-dimensional permutation-invariant subspaces other than Sym|αβγ〉 and Anti|αβγ〉. Suppose that we consider an analogue of vector (3.4) containing all permuted triples |ijk〉 and thus potentially permutation-invariant. In order to make this new vector orthogonal to Sym|αβγ〉 and Anti|αβγ〉, the coefficients of its six components have to form the sixth roots of unity: 1, ω, ω2, ω3, ω4, ω5, where ω = eπi/3. But this set forms the cyclic group Z6 which is not isomorphic with the permutation group S3; hence permutations applied to the new vector will generally not produce the same vector.
- 14.
A more general formula covering cases (5.12) and (5.13) can be found in Peres (2002, p. 134), however with an error (switched symbols Φ± and Ψ± in (5.59)).
- 15.
For Huggett the existence of non-permutation-invariant observables acting in spaces of states for a given type of paraparticles is proof that paraparticles can be discerned by properties. He insists that operators acting on allowed spaces (i.e. such that they do not take us outside a given allowed space) are perfectly admissible as representations of physical properties, even when they are not symmetric (Huggett 2003, p. 245ff). However, Peres disagrees with that, maintaining that all operators pertaining to “indistinguishable” particles, whether bosons, fermions or paraparticles, should be invariant under relabeling of particles (Peres 2002, p. 135).
- 16.
Observe, incidentally, that Peres, like virtually all physicists, continues talking about distinguishable properties of particles of the same type (such as the location on the Moon versus the location in a lab), in spite of the symmetry of the joint state of these particles and the ensuing Indiscernibility Thesis. We will propose a firm theoretical foundation for this practice in Chap. 5 and subsequent chapters.
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Bigaj, T. (2022). The Source of the Symmetrization Postulate. In: Identity and Indiscernibility in Quantum Mechanics. New Directions in the Philosophy of Science. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-74870-8_3
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