Abstract
We discuss the case against Factorism, which is the standard assumption in quantum mechanics that the labels of the \(\otimes\)factor Hilbertspaces in directproduct Hilbertspaces of composite physical systems of similar particles refer to particles, either directly or descriptively. We distinguish different versions of Factorism and argue for their truth or falsehood. In particular, by introducing the concepts of snapshot Hilbertspace and Schrödingermovie, we demonstrate that there are Hilbertspaces and \(\otimes\)factorisations where the labels do refer, even descriptively, to similar particles, which renders them probabilistically absolutely discernible.
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1 Factorism and Part States
Consider \(N\in {\mathbb {N}}_{+}\) particles, labelled by numerals ‘1’,‘2’, \(\ldots\),‘N’. In quantum mechanics (QM), every possible pure state of particle \(j\in \{1,2,\ldots ,N\}\) is mathematically represented by a Hilbertvector of norm equal to 1, up to a global phase factor (\(\mathrm {e}^{\mathrm {i}\theta }\), \(\theta \in [0,2\pi )\)), belonging to Hilbertspace \(\mathcal {H}_{j}\). The Hilbertspace \(\mathcal {H}\) of a composite system \(\textsf {S}^N\) of N particles is (a subspace of) the directproduct Hilbertspace of the N \(\otimes\)factor Hilbertspaces:
A vector in \(\mathcal {H}^{N}\) is symmetric iff it is invariant under every permutation of the labels, and antisymmetric iff it is invariant under every even permutation, and invariant up to a factor \(1\) for every odd permutation; and similarly for statistical (or density) operators in \(\mathcal {S}(\mathcal {H})\), which represent mixed states. According to the Symmetrisation Postulate of QM, for composite systems of similar fermions (i.e. having the same superselected, or stateindependent, properties, like mass, charge, spin magnitude and baryon number), \(\mathcal {H}\) is the antisymmetric subspace of \(\mathcal {H}^{N}\), denoted by \(\mathcal {H}_{}\), and for similar bosons, \(\mathcal {H}\) is the symmetric subspace of \(\mathcal {H}^{N}\), denoted by \(\mathcal {H}_{+}\). All \(\otimes\)factor Hilbertspaces must then be chosen the same: \(\mathcal {H}_{j}=\mathcal {H}_{k}\) for every \(j,k\in \{1,2,\ldots ,N\}\).^{Footnote 1}
The permutationoperators act by definition on the labels of the \(\otimes\)factor Hilbertspaces, e.g. for \(N=3\) and permutation 321 of 123:
One can suppress the labels in the vectors in (2) and let the typographical order do the labelling:
Since \(u\rangle _{1},w\rangle _{1}\in \mathcal {H}_{1}\), \(v\rangle _{2},\in \mathcal {H}_{2}\), and \(w\rangle _{3},u\rangle _{3}\in \mathcal {H}_{3}\), and Hilbertspace \(\mathcal {H}_{1}\) is associated with particle 1, etc., what could be more natural than to assume that the labels also refer to the particles? Indeed, nothing could be more natural. In fact, we opened this article by introducing the labels as names of particles and then used them too as labels of the \(\otimes\)factor Hilbertspaces.^{Footnote 2} This assumption seems to have been first unearthed explicitly and turned into a subject of serious thought by Lubberdink in her msthesis of 1998, parts of which were published 13 years later with her supervisor Dieks; independently it was implicitly unearthed by Huggett and Imbo (2009), and essentially by Saunders in 2006; the assumption has been baptised insipidly Factorism by Caulton, first in his PhDthesis of 2011.^{Footnote 3} Insipidly, we follow suit:
Factorism. Given a Hilbertspace \(\mathcal {H}\) of a composite system \(\textsf {S}^N\) of N given similar parts, representing the possible physical states of \(\textsf {S}^N\), and given a directproduct factorisation of \(\mathcal {H}\) into other Hilbertspaces. Then the labels of these given \(\otimes\)factor Hilbertspaces refer to the N parts either directly (Direct Factorism) or descriptively (Descriptive Factorism).
A number of systematic remarks on Factorism follow next.
First, we must realise that Factorism can only be judged true or false until and unless \(\mathcal {H}\) has been specified and a \(\otimes\)factorisation of \(\mathcal {H}\) has been specified. When one uses symbols like ‘\(\mathcal {H}\)’ and ‘\(\mathcal {H}_{j}\)’, then nothing has been specified, because these are variables ranging over the class of all and only Hilbertspaces. Then Factorism does not express a proposition that can be true or false. Factorism can be made independent of such specifications and then expressing a proposition, by quantifying the twofold dependence away universally (\(\mathbf {\forall }\)Factorism) or existentially (\(\mathbf {\exists }\)Factorism), as follows:
\(\mathbf {\forall }\)Factorism. For every Hilbertspace \(\mathcal {H}\) of a composite system \(\textsf {S}^N\) of N given similar parts and for every directproduct factorisation of \(\mathcal {H}\) into other Hilbertspaces, the labels of these given \(\otimes\)factor Hilbertspaces refer to the N parts either directly (Direct \(\mathbf {\forall }\)Factorism) or descriptively (Descriptive \(\mathbf {\forall }\)Factorism).
\(\mathbf {\exists }\)Factorism. For some Hilbertspace \(\mathcal {H}\) of a composite system \(\textsf {S}^N\) of N given similar parts and for some directproduct factorisation of \(\mathcal {H}\) into other Hilbertspaces, the labels of these given \(\otimes\)factor Hilbertspaces refer to the N parts either directly (Direct \(\mathbf {\exists }\)Factorism) or descriptively (Descriptive \(\mathbf {\exists }\)Factorism).
It will be useful to define even a fourth thesis:^{Footnote 4}:
AntiFactorism. For every Hilbertspace \(\mathcal {H}\) of a composite system \(\textsf {S}^N\) of N given similar parts and every directproduct factorisation of \(\mathcal {H}\) into other Hilbertspaces, the labels of these given \(\otimes\)factor Hilbertspaces neither refer to the N parts directly (Direct Factorism) nor descriptively (Descriptive Factorism).
These three theses express distinct propositions, which are logically related. If \(\mathbf {\forall }\)Factorism is true, then \(\mathbf {\exists }\)Factorism is true, and AntiFactorism is false. If \(\mathbf {\exists }\)Factorism is true, then nothing follows about \(\mathbf {\forall }\)Factorism, it can be true and it can be false; but then AntiFactorism again is false. If \(\mathbf {\forall }\)Factorism is false, then both \(\mathbf {\exists }\)Factorism and AntiFactorism can be true or can be false. If AntiFactorism is true, then both \(\mathbf {\forall }\)Factorism and \(\mathbf {\exists }\)Factorism are false. If AntiFactorism is false, then \(\mathbf {\exists }\)Factorism is true and \(\mathbf {\forall }\)Factorism can be true or false. It will transpire that \(\mathbf {\exists }\)Factorism is true, \(\mathbf {\forall }\)Factorism is false, and hence AntiFactorism is also false.
Notice that for composite systems of dissimilar parts, the labels of the \(\otimes\)factors do refer unproblematically to these dissimilar parts. Which is why all versions of Factorism mention explicitly that the parts must be similar.
Secondly, it is rarely clear which version of Factorism is criticised by the authors listed above. Their generic use of symbols like ‘\(\mathcal {H}\)’ and ‘\(\mathcal {H}_{j}'\) suggest they talk about an arbitrary Hilbertspace, which makes it logically equivalent to talking about all Hilbertspace (by the introductionrule of ‘\(\mathbf {\forall }\)’). This would be AntiFactorism and therefore they seemed to believe to have established the truth of AntiFactorism. The context however usually reveals they have been considering a specific Hilbertspace, notably Dieks and Lubberdink (2011), and a specific \(\otimes\)factorisation, which means that all they have established is that there is a Hilbertspace and there is a \(\otimes\)factorisation of which the labels do not refer to the particles, in other words, they have established that \(\mathbf {\forall }\)Factorism is false by producing a counterexample. This leaves the truth of both \(\mathbf {\exists }\)Factorism and AntiFactorism wide open. Again, we shall conclude the falsehood of AntiFactorism and thereby the truth of \(\mathbf {\exists }\)Factorism.
Thirdly, we must also realise that every version of Factorism is an assumption, because it does not follow from ‘the mathematics’ or the Von Neumann postulates of QM. Like postulating that certain Hilbertvectors represent physical states and certain operators represent physical magnitudes, Factorism is an assumption that tells us that certain mathematical entities, the labels (for which standardly positive natural numbers are used), besides naming \(\otimes\)factor Hilbertspaces and naming operators acting in them (generically operator \(A_{j}\) acting in \(\mathcal {H}_{j}\)), are also names of the particles. Physical significance is attributed to these labels—and explicity denied by AntiFactorism.
Fourthly, as Friebe has pointed out, connecting Factorism to the philosophy of language, that standardly the labels refer directly to the particles, they are senseless names, in contrast to descriptive names, which refer descriptively by Russellian definite descriptions.^{Footnote 5} Whence the difference between Direct and Descriptive Factorism. In order to refer descriptively to a constituent particle of a physical system, the particle must be absolutely discerned from the other constituent particles, which means it must possess at least one property that all other particles lack. In the case under consideration, of physical systems composed of similar particles, descriptive reference seems impossible because all constituent particles share all their properties and seem therefore absolutely indiscernible; direct reference is then the only available option to refer. Of course, one can discern the particles absolutely by means of labels (the particle having label ‘1’), but since these senseless labels lack any physical significance, such discernment is physically insignificant. The particles remain physically, or empirically, absolutely indiscernible. (In fact, what is permitted to discern and what is forbidden to discern in the context of QM—and beyond—has recently been scrutinised.^{Footnote 6}) When probability measures occur in the relations, one speaks of probabilistic discernment.
Do the constituent parts of a composite physical system generally also have states? Call the answer in the affirmative:
Part States. All parts of a composite physical system have states.
As soon as an answer in the affirmative has been given, the next question is: what is the state of the parts, and how does it relate to the state of the composite system? Are the states of similar particles different? Before we address these questions, let us first see what arguments are around in favour of adopting Part States.
The most general conception of a physical state is the mixed one, as Von Neumann proposed, and mixed states are represented by statistical operators in the convex set \(\mathcal {S}(\mathcal {H}_{\pm })\) (positive, selfadjoint, trace class\(\,1\)); the expectationvalue of a physical magnitude represented by symmetric operator A when the physical system is in state W is provided by the celebrated traceformula:
By choosing for A the members of its spectral resolution, formula (4) yields a probabilitymeasure for Ameasurement outcomes.
\(1.\) One reason in favour of Part States is that Part States needs no reason in QM because in QM it follows from the State Postulate of QM: (i) the members of \(\mathcal {S}(\mathcal {H})\) represent the possible physical states of a physical system with which \(\mathcal {H}\) is associated, and (ii) the members of \(\mathcal {S}(\mathcal {H}_{1}\otimes \mathcal {H}_{2}\otimes \cdots \mathcal {H}_{N})\) represent those of a composite system of N parts. Since the parts of the lastmentioned physical system are physical systems too, with which the \(\otimes\)factor Hilbertspaces are associated, it follows from (i) that the possible states of part j are represented by members of \(\mathcal {S}(\mathcal {H}_{j})\). Yet one may weaken the State Postulate to (ii), and leave it open whether parts have states too, or even deny that parts have states. In that case, Part States does not follow from the State Postulate, and we may now legitimately request a reason for Part States.
\(2.\) The standard reason for Part States goes as follows. In principle we can perform a measurement of any physical magnitude pertaining to a part, viz. the position of a single particle, represented by a symmetrised oneparticle positionoperator. Repeated measurements test a singleparticle probabilitymeasure. Since probabilitymeasures of physical magnitudes pertaining to the composite system are generated by some statistical operator \(W\in \mathcal {S}(\mathcal {H}_{\pm })\) and some symmetric operator A acting in \(\mathcal {H}_{\pm }\) (W is guaranteed to exist by Gleason’s Theorem), we also want singleparticle probabilitymeasures to be generated by some statistical operator \(W_{j}\in \mathcal {S}(\mathcal {H}_{j})\) and some operator \(A_{j}\) acting in \(\mathcal {H}_{j}\) (\(W_{j}\) is also guaranteed to exist by Gleason’s Theorem). But which one? As Van Fraassen has pointed out, all we need to obtain Von Neumann’s choice that \(W_{j}\) is the partial trace of W is to endorse the eminently reasonable requirement that the expectationvalue of \(A_{j}\) acting in \(\mathcal {H}_{j}\) when particle\(\;j\) is in state \(W_{j}\) equals the one of the trivially extended \(A_{j}\) to act symmetrically in \(\mathcal {H}_{\pm }\) when the composite system is in state \(W\in \mathcal {S}(\mathcal {H}_{\pm })\):^{Footnote 7}
For two particles, A is \({\frac{1}{2}}(A_{1}\otimes \textit{1}\,_{\!2}+\textit{1}\,_{\!1}\otimes A_{2})\), etc.
Notice that in both reasons, \(\mathbf {\forall }\)Factorism seems to have been taken for granted.
Next it is elementary to show that the partial traces of similar fermions are identical, and for similar bosons too, so that the singleparticle states cannot be used to discern the particles absolutely, and hence cannot be used to refer to the particles descriptively in what would have been a physically significant manner.^{Footnote 8} For fermions, the states are always improper mixtures; for bosons, pure states and improper mixtures are possible, never proper mixtures. Thus the similar particles remain absolutely indiscernible, even after Part States has been embraced. They cannot be individuated and therefore are not individuals; they are indiscernibles. The adoption of Part States has brought no change in the situation that according to QM, similar particles are indiscernibles. Moreover, Direct Factorism is the name of the reference game of similar particles in QM. Descriptive Factorism is out.
In order to prevent that an operator representing a physical magnitude takes Hilbertvectors outside \(\mathcal {H}_{\pm }\), it must be symmetric, which is to say that it must commute with every permutation operator. This is known as the Indistinguishability Postulate, which is not a postulate because it follows from the Symmetrisation Postulate.^{Footnote 9} That symmetrised operators must be used when considering similar particles is wellknown. Less wellknown is that when we want to measure a physical magnitude pertaining to a single particle, one must distinguish, in case of two particles, between (i) whether a single particle has a property and the other does not have it, and (ii) whether at least one particle has it. Bigaj (2015b, 193–195) explains how and why.
Some have argued that the similar particles can be discerned by relations in physically significant ways, which turns the similar particles into relationals, in contrast to indiscernibles and individuals.^{Footnote 10} The first criticism against this relational discernment was that it is circular; this criticism has not stood up to scrutiny.^{Footnote 11} The second criticism is that the discerning relations in some instances employ operators that do not obey the Indistinguishability Postulate; hence no physical significance is bestowed on the discerning relations by the operators employed in defining these relations.^{Footnote 12} Repair jobs of relational discernment have been mounted; some repair jobs have been succesful for some but not all classes of states, another repair job has been succesful for all states but is probabilistic in nature.^{Footnote 13} Leibniz’s Principle of the Identity of Indiscernibles is a metaphysical principle that is at stake in this debate. The proponents of relational discernibility have argued that the sufficient condition for identity of sharing all properties must be extended to include also relations, in which case the principle is obeyed rather than violated by QM. The debate about relational discernibility and Leibniz’s principle continues but we park it here. For even if similar particles are relationals, they are not individuals, not absolutely discernible, and Direct Factorism then seems the only manner to refer to the particles.
We next turn to the arguments against Factorism (Sects. 2, 5), which we shall counter (Sects. 3, 5). Further, we mention a few things that are just nontrivial enough to be worthy to be pointed out at all (Sect. 4, which can be omitted) and we end with concluding remarks (Sect. 6).
2 Factorisable, Tangled and Entangled States
Consider a composite system of a proton and an electron with Hilbertspaces \(\mathcal {H}_{1}=L^{2}({\mathbb {R}}^{3})\) and \(\mathcal {H}_{2}={\mathbb {C}}^{2}\) (which is admittedly a peculiar combination). This system can be in some productstate:
The particle labelled ‘1’ can now be descriptively referred to as ‘the particle having stateindependent properties mass \(m_{\text {p}}\), charge \(+1\) and spinmagnitude 1/2’, and mutatis mutandis for ‘2’. The particles can also be descriptively referred to by their states: particle\(\;1\) is the particle in the state represented by wavefunction \(\psi\), and particle\(\;2\) is the particle in the state represented by the columnvector in (6). In superpositions, which prohibit the ascription of properties corresponding to magnitudes whose eigenvectors are members of the basis in which the superposition is expressed, neither statedependent properties nor states of parts can be used for descriptive reference. But the stateindependent properties can always be used to do the job of descriptive reference. These particles are absolutely discernible, they can be discerned in physically significant ways, they are individuals, and they can be individuated by their stateindependent properties, which they always possess. For composite systems of dissimilar particles, Descriptive Factorism holds good. Direct Factorism holds good too, but has become otiose.
There are three arguments against Factorism around, which all pertain to composite systems of similar particles:

The Argument from Tangled States (current section).

The Argument from Entangled States (current section).

The Argument from Measurement (Sect. 5).
Before we exposition these arguments, first some preparations.
Since similar particles cannot be discerned by: their stateindependent properties, by their physically insignificant senseless labels, or by their states if they have them (their states are all the same), the only way to discern them is by means of their statedependent properties (or relations). Here the troubles begin. Productstates, like \(u\rangle _{1}\otimes v\rangle _{2}\), cannot be antisymmetric, so turning the directly referring labels into descriptive ones by means of the states is not on.^{Footnote 14} Productstates can be symmetric, but only when the \(\otimes\)factors are alike, as in: \(u\rangle _{1}\otimes u\rangle _{2}\), which means that both particles are in exactly the same state \(u\rangle\) and hence discernment by means of the states is not on.
The inquiries into entanglement by Ghirardi et al. (2002) c.s. has however led to the view that there are nonfactorisable states in which particles can be discerned by means of descriptive reference.^{Footnote 15} They call nonfactorisable states that result from antisymmetrising a productstate ‘not entangled’; we shall call them tangled. Call a nonfactorisable state that does not result from (anti)symmetrising some productstate entangled. The nonfactorisable vectors then subdivide in tangled and entangled ones. One good reason to draw this distinction in the nonfactorisable states is that only entangled states, when considering spin and position, lead to a breach of a Bellinequality, whereas tangled states do not lead to a breach.^{Footnote 16} When in 1935, Schrödinger famously wrote about what he called ‘entanglement’: “I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought”, one can only endorse this when by ‘entanglement’ is meant how Girardi et al. characterised it, to the exclusion of the nonfactorisable tangled states, which do not warrant a departure from classical lines of thought. (But see the Envoi in Sect. 7.)
For the sake of clarity and future reference, here are three running examples of Hilbertvectors of each type, for two similar spin 1/2fermions, with Paulispinor Hilbertspace \(L^{2}({\mathbb {R}}^{2})^{2}\) as the singleparticle statespace (thus including all degrees of freedom: position, momentum, spin):
Wave functions \(L;\,t\rangle ,R;\,t\rangle \in L^{2}({\mathbb {R}}^{3})\) are wavepackets moving in opposite directions (called ‘Left’ and ‘Right’) in configuration space \({\mathbb {R}}^{3}\):
and
with dispersion function \(\omega (k)=\hslash k^{2}/2m\), wavevector \({\mathbf {k}}=\mathbf{p} /\hslash =m \mathbf{v }/\hslash\), and length \(\;k=\Vert {\mathbf {k}}\Vert\). Velocity \(\mathbf{v}\) is the velocity of the peak of the wavepacket in configurationspace—not the velocity of the particle. Here are the Paulispinors:
and mutatis mutandis for \(R,\!\uparrow _{z};t\rangle\) and \(R,\!\downarrow _{z};t\rangle\).
The Argument against Descriptive Factorism from Tangled States. As we mentioned above, factorisable vectors like \(\Phi (t)\rangle\) do not represent possible physical states of a similar particles system. Ghirardi c.s. submit that the tangled vector \(\Theta (t)\rangle\) represents a state such that one particle has zspin up and is moving to the left (LeftzUp) and the other particle has zspin down and is moving to the right (RightzDown).^{Footnote 17} For dissimilar particles, such a state is represented by \(\Phi (t)\rangle\), but the Symmetrisation Postulate forbids this and makes us antisymmetrise \(\Phi (t)\rangle\), which yields \(\Theta (t)\rangle\). This interpretation of \(\Theta (t)\rangle\) contravenes the standard Strong Property Postulate (aka under the somewhat misleading name ‘EigenstateEigenvalue Link’^{Footnote 18}), according to which a physical system in pure state \(\psi \rangle\) has quantitative property \(\langle A,a\rangle\) iff \(A\psi \rangle =a\psi \rangle\).^{Footnote 19} Hilbertvector \(\Theta (t)\rangle\) is neither in an eigenstate of any spectral projector of the positionoperator (wavepackets have infinite tails) nor in an eigenstate of \(\sigma _{z}\), and then can neither be localised in any spatial region nor possess a zspin property \(\langle \sigma _{z},\pm 1\rangle\). Particles can have quantitative properties even when not in eigenstates, say Ghirardi et al., provided their nonfactorisable state is tangled. Descriptive reference becomes a viable option again.^{Footnote 20}
For the Factorist who wants to endorse this interpretation, the following problem arises. Was the particle initially directly referred to by label ‘1’ LeftzUp or RightzDown? The Factorist cannot answer: it has to be one of them, because we all agree there are two particles. The descriptions LeftzUp and RightzDown are in the tangled state associated to two vectors, each one in a different Hilbertspace, and therefore associated to both labels ‘1’ and 2’.
So it seems there is strong case that for tangled states, the Hilbertspace labels do not refer to particles, and for those states, Direct as well as Descriptive Factorism fails.
The Argument against Factorism from Entangled States. The move to descriptive reference is unavailable for the entangled state \(\Psi (t)\rangle\) (7), which is empirically distinguishable from the tangled state \(\Theta (t)\rangle\), for there is no correlation anymore in the terms between spin and space. Both Alice (at the left wing of the celebrated Einstein–Podolsky–Rosen–Bohm–Bell experiment) and Bob (at the right wing) have a probability of 1/2 to find zspin up and to find zspin down, whereas in the tangled state \(\Theta (t)\rangle\), Alice has nearly probability equal to 1 to find spin up (LeftzUp) and equally Bob to find spin down (RightzDown).^{Footnote 21} Spin and space now have become entangled. Direct reference returns. One can maintain that the Hilbertspace labels refer to the particles and that these senseless names cannot be filled with physical significance by means of physically significant descriptions as in the tangled case. Descriptive Factorism is not on. Yet this implies that Direct Factorism with its senseless, referring labels remains the only live option to refer.
But perhaps descriptive reference is on after all. When in the tangled state \(\Theta (t)\rangle\) we can have LeftzUp and RightzDown, why can we not have in the entangled state \(\Psi (t)\rangle\) Leftandnozspinproperty and Rightandnozspinproperty? We can have that.
We can calculate the probability that Alice and Bob find a particle in their detector, while not measuring its spin; they then perform a position measurement. Here are the Born positionprobability densities, which ‘move’ in ordinary space (represented by \({\mathbb {R}}^{3}\)); one is moving towards Alice:
and the other one is moving towards Bob:
where \(1_{\{{\mathbf {r}}\}}\) is the characteristic function of singletonset \(\{{\mathbf {r}}\}\), so that \(1_{\{{\mathbf {r}}\}}(\mathbf {q})=1\) if \(\mathbf {q}={\mathbf {r}}\), and \(1_{\{{\mathbf {r}}\}} (\mathbf {q})=0\) if \(\mathbf {q}\ne {\mathbf {r}}\).^{Footnote 22} We have again two absolutely discernible particles, discerned by their distinct positionprobability densities, \(\rho _{L}\) and \(\rho _{R}\), respectively. The particles have become individuals.
Dieks and Lubberdink (2011) have indicated that this way of looking at entangled states—and tangled states, if not all states whenever warranted—is needed for the classical limit of QM. The peaks of the positionprobabilities in space, provided we end up with N singlepeaked positionprobabilities in space for Nparticle systems, follow classical trajectories according to Ehrenfest’s Theorem, because the peaks move with a velocity in space equal the expectationvalue of the velocityoperator in such cases.
The situation with the entangled state thus has become the same as with the tangled state. Both position probabililtydensities involve both labels, manifest in (11) and in (12) by ‘\(\mathbf {q}_{1}\)’ and ‘\(\mathbf {q}_{2}\)’. So just as in the tangled case, the Hilbertspace labels do not refer to particles. Direct as well as Descriptive Factorism fails.
There is however one caveat. The wavepackets of noninteracting particles, as in the EPRBcase, spread rather quickly. Within no time, the positionprobability densities are nearly uniform. For example, when we consider electrons and a Gaussian wavepacket having a width equal to the Bohr radius (the size of an Hatom in its ground state: \(10^{11}\,\)m), then after one second, the width has become:
which is about three times the Lunar distance. The probability to find the electron at 1 second beyond the moon is 1/3. The situation of absolute discernment is over in the wink of an eye.
Cold comfort for the Factorist, because a situation of nearly uniform wavepackets is not going to help Factorism to regain its long reign of dominance, albeit shifting Hilbertspaces. So Factorism rest in peace after all? Not so quick. All considerations above are about a specified Hilbertspace, the antisymmetric subspace of infinitedimensional Paulispinor Hilbertspace:
But what if we specify a different Hilbertspace? We shall see what then happens in the next section.
3 Snapshot HilbertSpace and SchrödingerMovies
We can bury \(\mathbf {\forall }\)Factorism with appropriate honours, that much should be evident by now. But how about \(\mathbf {\exists }\)Factorism?
We now construct a 4dimensional proper subspace of Paulispinor space (14) with descriptively referring \(\otimes\)factor Hilbertspace labels. During the short timeinterval where the criticism of Factorism has bite, the pairs of wavefunctions are approximately orthogonal:
Now we consider the directproduct Hilbertspace spanned by the following tangled vectors:^{Footnote 23}
We \(\otimes\)factorise this snapshot Hilbertspace \(\mathcal {H}(t)\) into \(\mathcal {H}_{\mathrm {l}}(t)\) and \(\mathcal {H}_{\mathrm {r}}(t)\) such that:
where \(\,\!\uparrow _{z};t\rangle _{\mathrm {l}}\in \mathcal {H}_{\mathrm {l}}(t)\,\) and \(\,\!\uparrow _{z};t\rangle _{\mathrm {r}}\in \mathcal {H}_{\mathrm {r}}(t)\). Then we have:
The history of the twofermion system is some curve in \(L^{2}({\mathbb {R}}^{6})^{4}\). The continuous sequence of 4dimensional snapshot Hilbertspaces \(\,t\mapsto \mathcal {H}_{\mathrm {l}}(t)\otimes \mathcal {H}_{\mathrm {r}}(t)\), which we call the Schrödingermovie, envelops this history.^{Footnote 24} In this movie, the singleparticle labels ‘\(\mathrm {l}\)’ and ‘\(\mathrm {r}\)’ can be taken as descriptive names: particle \(\mathrm {l}\) is the particle moving to the left in space (Left) and \(\mathrm {r}\) is the particle moving to the right in space (Right), but neither particle possesses a zspin property \(\langle \sigma _{z},\pm \rangle\).^{Footnote 25} So when it said that we are dealing with ‘two particles in the singlet state moving in opposite directions’, then this description does not fit the Bohm singlet state:
because the spatial degrees of freedom are absent from (19), but these words do fit the spinspace entangled state \(\Psi (t)\rangle\) (7) in the snapshot basis:
When we watch the tangled \(\Theta (t)\rangle\) in the Schrödingermovie, it factorises:
which has singleparticle Hilbertspace labels that we take to refer descriptively: \(\mathrm {l}\) is LeftzUp and \(\mathrm {r}\) is RightzDown.
So neither the argument against Factorism from Tangled States nor the one from Engtangled States is as decisive as their proponents believe: they wound \(\mathbf {\forall }\)Factorism lethally, but leave \(\mathbf {\exists }\)Factorism scatheless. In fact, for the case of two spin1/2 fermions, we have proved \(\mathbf {\exists }\)Factorism.
Before we turn to the third and last argument against Factorism, we first want to make a few remarks inspired by our Schrödingermovie.
4 Intermezzo: Parts and Degrees of Freedom
This section contains a few explanatory remarks about QM in general, which can be passed over. An arbitrary vector in the Paulispinor Hilbertspace of one electron can be expanded based on basisvectors \(\alpha _{j}\in L^{2}({\mathbb {R}}^{3})\) (\(j\in {\mathbb {N}}_{+}\)):
When we now write:
it is obvious that we can write Paulispinor space as the direct product \(L^{2}({\mathbb {R}}^{2})\otimes {\mathbb {C}}^{2}\). This shows that the physical significance of ‘\(\otimes\)’ is ambiguous. Usually it marks the combination of Hilbertspaces pertaining to the different subsystems of a composite system. But here we have a single electron, which has no parts. In this case, ‘\(\otimes\)’ marks a separation of (spatial and spin) degrees of freedom.
For two spin1/2 particles, we then can write (the antisymmetric subspace of) Paulispinor Hilbertspace (14) as follows:
The physical significance of ‘\(\otimes\)’ in the left expression is one of different composing parts, in the right expression one of separation of different degrees of freedom, and in the middle expression both kinds of physical significance occur.
Another thing to notice about ‘\(\otimes\)’ is that \(\otimes\)factorisations are not unique, which has as a consequence that by staring at a \(\otimes\)factorised Hilbertspace, we will not get a clue what the modelled physical system is and even not about how many particles compose it. For example, we have:
which is a state space of two particles in 3 spatial dimensions, and of three particles in the plane (\({\mathbb {R}}\)), and of six particles on a line (\({\mathbb {R}}\))—and even of one particle in a 6dimensional Euclidean space (\({\mathbb {R}}^{6}\)). Moreover, all these Hilbertspaces are also isomorphic to \(L^{2}({\mathbb {R}})\), pertaining to a single particle on a line. Moral: to specify a Hilbertspace, which represents the set of all possible physical states of a physical system, does next to nothing to specify that physical system.
The \(\otimes\)factorisation of spin and space in (24) does not mean that spin and space also \(\otimes\)factorise in every state. But in the case of the antisymmetric subspace they do. Here is again the spacespin singlet (7), in degreeoffreedomfactorised state:
A vector slightly deviating from antisymmetry, and therefore not a member of the antisymmetric statespace, is the following one in snapshot Hilbertspace, which deviates slightly from \(\Psi (t)\rangle\) (20):
This vector represents a state of two dissimilar particles. The vector does not \(\otimes\)factorise in a spin and a spatial part. In the grand Hilbertspace (24), this vector looks as follows:
Hence no snapshot Hilbertspace for dissimilar particles ensues—which is not bad news for Factorists because for dissimilar particles, even \(\mathbf {\forall }\)Factorism rules.
5 Measurement and Locality
Friebe (2014) considers the tangled state \(\Theta (t)\rangle\) (7). Bob performs a joint measurememt on one particle of yspin and of a projector in the spectral resolution of the positionoperator, that projects on the volume in space occupied by the yspin detector of Bob (\(V_{R}\subset {\mathbb {R}}^{3}\)). (This joint measurement of spin and position is possible because the corresponding operators commute.) Alice does not perform any measurement. The symmetric operator that represents Bob’s measurement is this one:
We further assume that Bob measures at a time \(t=\tau\) when the bulk of the positionpropability of \(R;\tau \rangle\) is at \(V_{R}\), so that \(P^\mathbf{Q }R;\tau \rangle \approx R;\tau \rangle\), which implies that the probability to detect one particle at \(t=\tau\) at \(V_{R}\) is nearly 1. Since the eigenstate of yspin down is a superposition of eigenstates of zspin up and zspin down, the probability to find yspin down at \(V_{R}\) equals 1/2.
Suppose Bob finds yspin down and detects one particle at \(V_{R}\). State \(\Theta (t)\rangle\) then collapses to:
where \(R,\!\downarrow _{y};\tau \rangle\) is the Paulispinor localised at \(V_{R}\) and being an eigenvector of yspin with eigenvalue \(1\). The collapsed state \(\Theta (\tau )\rangle\) (30) remains a tangled state. There was a particle LeftzUp before the measurement and there remains one after the measurement. The particle RightzDown, on which Bob has performed the measurement, is now RightyDown.
Next Friebe points out that if Factorism is correct (and Part States is assumed), then the postmeasurement states of both particles are the partial traces of (30), and they are the same. Furthermore, their state is different from their premeasurement state. The change of state of one particle is due to Bob’s measurement of yspin and location in \(V_{R}\) of this particle (labeled ‘1’ or ‘2’ but one cannot know which), whereas Alice has left the other particle alone. Then why has its state changed? In case of an entangled state, we would, versed in QM as we are, have expected this, but \(\Theta (\tau )\rangle\) is a tangled state.^{Footnote 26}
Give up direct reference, and thereby Direct Factorism, embrace Ghirardi’s interpretation of tangled states, which entails giving up Descriptive Factorism too, and the problem evaporates.
When we move to snapshot Hilbertspace (see Sect. 3), the tangled state \(\Theta (\tau )\rangle\) (30) \(\otimes\)factorises and the labels of the \(\otimes\)factor Hilbertspaces can be provided with descriptions. Descriptive Factorism with specified snapshot Hilbertspace has been vindicated, and this proves \(\mathbf {\exists }\)Factorism.
6 Concluding Remarks
When we enter the noisy movie theatre of the grand Hilbertspace with its infinitely many dimensions (14), (24), Factorism plays the part of the whipping boy in the movie Composite Physical Systems of Similar Particles, for the tangled as well as the entangled cases. But when we watch the Schrödingermovie in a continuous sequence of snapshot Hilbertsubspaces of four cosy dimensions, Factorism is the indomitable hero: the labels of the \(\otimes\)factor Hilbertspaces of snapshot Hilbertspace can be taken to play descriptive roles. Descriptive reference and \(\mathbf {\exists }\)Factorism vindicated. In both Hilbertspaces we can calculate the relevant probabilities of the position and spin degrees of freedom.
Finally, after having kissed Factorism goodbye, Caulton (2012; 2018a, b) devised the following construction. For the sake of simplicity, we stick to a composite system of two similar particles, with Hilbertspace
Let \(E_{\alpha },E_{\beta }\in \mathcal {L}(\mathcal {H})\) be two orthogonal projectors (members of the lattice), which will individuate the two particles—thus the particles will become absolutely discernible. Consider the following subspace:
obtained by letting a symmetric operator act on \(\mathcal {H}^{2}_{\pm }\), thus guaranteeing that its range lies inside \(\mathcal {H}^{2}_{\pm }\). One particle always has a state in subspace \(\mathcal {H}_{\alpha }\equiv E_{\alpha }[\mathcal {H}]\sqsubset \mathcal {H}\) and the other particle \(\mathcal {H}_{\beta }\equiv E_{\beta }[\mathcal {H}]\sqsubset \mathcal {H}\).
The state of the composite twoparticle system then is in \(\mathcal {H}_{\alpha }\otimes \mathcal {H}_{\beta }\). The labels of these \(\otimes\)factor Hilbertspaces then can be taken as descriptive names of the particles: the particle with state in \(\mathcal {H}_{\alpha }\) and the particle with state in \(\mathcal {H}_{\beta }\). Caulton ends up with nothing less than two absolutely discernible particles—and even Leibniz’s Principle of the Identity of Absolute Indiscernibles is vindicated uno tenore.
Ironically, Caulton began by rejecting Factorism (in fact \(\mathbf {\forall }\)Factorism) and intended to devise an ‘antifactorist’ alternative \(\otimes\)factorisation of the Hilbertspace of the composite system, only to end up with a rigorous proof of \(\mathbf {\exists }\)Factorism and a refutation of AntiFactorism. Smile, please.
7 Envoi
We baptised nonfactorisable states that result from antisymmetrising a productstate as tangled (p. 8). Ghirardi et al. claimed that only entangled states lead to violations of Bell’s inequality, and thus that tangled states do not.^{Footnote 27} Yet take this product state:
Antisymmetrising this state yields:
This state then must be tangled, not entangled, according to Ghirardi et al., and therefore should not violate any Bell inquality. But \(\Xi (t)\rangle\) (34) does violate a Bell inequality and is very entangled. Were Ghirardi et al. mistaken to propound that, contra communis opinio, tangled states do not deserve to be called ‘entangled’ because they do not lead to a violation of some Bell inequality?
Notes
Stricto sensu isomorphic: \(\mathcal {H}_{j}\simeq \mathcal {H}_{k}\), for the \(\otimes\)factors being isomorphic will also do, even when they are not identical; since all complex (separable) Hilbertspaces of the same dimension are isomorphic, it is thus stricto sensu sufficient to require that the \(\otimes\)factor Hilbertspaces have the same dimension.
Standard in every textbook on QM and the overwhelming majority of papers on the subject of ‘identical’ (similar) particles. See e.g. CohenTannoudji et al. (1978), Butterfield (1993), French and Redhead (1988), Redhead and Teller (1992), French and Krause (2006), Saunders (2006a, b), Muller and Saunders (2008), Muller and Seevinck (2009), Ladyman et al. (2012).
Lubberdink (1998), Saunders (2006b), Huggett and Imbo (2009), Dieks and Lubberdink (2011), Caulton (2018b). Caulton currently prefers to call Direct Factorism “transcendental individuation”, harking back to a terminology of Redhead c.s. Caulton (2018a), Redhead and Teller (1992), French and Redhead (1988). The terminology of ‘transcendental individuation’ should not lead us to believe that a metaphysical issue is at stake here: the interpretation of the \(\times\)factor labels is a semantic issue.
The name of which has also been introduced by Caulton (2012, § 2.2): “Antifactorism is the doctrine that factor Hilbertspace indices are a gauge quantity.” These labels “do not represent anything physical”. When Caulton means here the \(\otimes\)factor Hilbertspaces of all \(\otimes\)factorisations of all Hilbertspaces, then this doctrine is the same as our AntiFactorism.
See Friebe (2014). To wit, Friebe defends descriptivism but not Descriptive Factorism.
Fraassen (1991, 199).
Fraassen (1992, 386), QMExclusion Corollary.
But notoriously not conversely. See Fraassen (1992, 381 ff), Huggett and Imbo (2009, § 3).
See references in the previous footnote, and Friebe (2014).
If \(u\rangle _{1}\otimes v\rangle _{2}= \Pi _{12}u\rangle _{1}\otimes v\rangle _{2}=v\rangle _{1}\otimes u\rangle _{2}\), then \(v\rangle\) or \(u\rangle\), or both, must be the zero vector; but the zero vector does never represent a physical state.
Ghirardi et al. (2002, 86).
Misleading, because the eigenvalueeq. connects eigenstates to eigenvalues, which is a mathematical connexion that does not involve properties. The Strong Poperty Postulate states a strong connexion between a physical system possessing quantitative properties and having a particular type of state.
Formulations encompassing continuous spectra and mixed states are available but glossed over.
About 25 years ago, Van Fraassen also presented, in the context of the modal interpretation of QM (which also renounces the Strong Property Postulate), a model of a composite system of fermions with individuation, that is, with descriptive reference, turning the fermions into absolutely discernible objects. Fraassen (1992, 427 ff.)
Nearly, because the position probabilities have infinite tails.
These positionprobability densities can also be obtained from the ones in configuration space by means of deltadistributions: \(\delta (\mathbf {q}_{1}{\mathbf {r}})\) and \(\delta (\mathbf {q}_{2}{\mathbf {r}})\).
This history is a solution of Schrödinger’s equation, whence the name.
The spatial meaning of the labels makes this come close to Friebe’s (2014) Kantian irreducible demonstratives.
To aggravate the problem, one can easily argue that we have a breach of EPRLocality here. Exercise.
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Thnx to Dennis Dieks, Tomasz Bigaj, Cord Friebe, Ronnie Hermens, Andrea Lubberding and two anonymous Referees for comments and suggestions.
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Muller, F.A., Leegwater, G. The Case Against Factorism: On the Labels of \(\otimes\)Factor HilbertSpaces of Similar Particles in Quantum Mechanics. J Gen Philos Sci 53, 291–306 (2022). https://doi.org/10.1007/s10838020095146
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DOI: https://doi.org/10.1007/s10838020095146