Informational approach to the quantum symmetrization postulate

A remarkable feature of quantum theory is that particles with identical intrinsic properties must be treated as indistinguishable if the theory is to give valid predictions in all cases. In the quantum formalism, indistinguishability is expressed via the symmetrization postulate (Dirac P 1926 Proc. R. Soc. A 112 661, Heisenberg W 1926 Z. Phys. 38 411), which restricts a system of identical particles to the set of symmetric states (‘bosons’) or the set of antisymmetric states (‘fermions’). However, the physical basis and range of validity of the symmetrization postulate has not been established. A well-known topological derivation of the postulate implies that its validity depends on the dimensionality of the space in which the particles move (Laidlaw M and DeWitt C 1971 Phys. Rev. D 3 1375–8, Leinaas J M and Myrheim J 1977 Il Nuovo Cimento B 37 1–23). Here we show that the symmetrization postulate can be derived by strictly adhering to the informational requirement that particles which cannot be experimentally distinguished from one another are not labelled. Our key novel postulate is the operational indistinguishability postulate, which posits that the amplitude of a process involving several indistinguishable particles is determined by the amplitudes of all possible transitions of these particles when treated as distinguishable. The symmetrization postulate follows by requiring consistency with the rest of the quantum formalism. The derivation implies that the symmetrization postulate admits no natural variants. In particular, the possibility that identical particles generically exhibit anyonic behavior in two dimensions is excluded.

The treatment of identical particles is one of the most remarkable features of quantum theory. In classical mechanics, identical particles can be treated as distinguishable since one can in principle follow the precise trajectory of each particle over time. However, identical particles in quantum theory must be treated as indistinguishable if the theory is to account for such phenomena as the chemical behaviour of multi-electron atoms.
Formally, the indistinguishability of the particles in a quantum system is expressed via the symmetrization postulate [1,2], which stipulates that a system consisting of a number of identical subsystems must be described by a state that is either symmetric or antisymmetric under any interchange of the state variables. For example, a wavefunction ψ x x x ( , 3 describing a state of three structureless identical particles must satisfy (1) (3) where π is any transposition, where the choice of sign is fixed for a given system. This postulate is responsible for the classification of all identical particles in nature into bosons and fermions, and is required for the understanding of a vast range of physical phenomena ranging from atomic and molecular structure to the spectrum of blackbody radiation.
However, the physical basis and range of validity of the symmetrization postulate remains uncertain. Shortly after positing the symmetrization postulate, Dirac noted that 'other more complicated kinds of symmetry are possible mathematically, but do not apply to any known particles' [5]. In the subsequent years, there have been numerous efforts to determine its physical basis and range of validity [3,4,[6][7][8][9][10][11][12][13][14]. However, the results are generally incomplete or inconclusive, and leave open the possibility that identical particles other than fermions and bosons may exist in nature. For example, the well-known topological approach [3,4] yields the symmetrization postulate in the special case of structureless particles moving in three or more spatial dimensions, but implies the breakdown of the symmetrization postulate in two spatial dimensions with so-called anyonic behaviour [4,15]. However, the topological argument is limited to structureless particles, and relies upon abstract assumptions such as the much-scrutinized assumption that point-particles cannot coincide and the conjectured generalization of Feynmanʼs sum rule for multiply-connected spaces [16]. Consequently, although anyons have found significant application [17,18], for example in the explanation of the fractional Hall effect, their status as elementary particles on a par with fermions and bosons remains uncertain.
A significant problem in most of the aforementioned attempts to understand the symmetrization postulate is their labelling of indistinguishable particles. Although labelling of entities that have no physical counterparts is a commonly accepted formal device in physical theories so long as the predictions are invariant under relabelling, the labelling of indistinguishable particles presents special difficulties [19][20][21][22], which has led to some approaches which explicitly attempt to avoid labelling of indistinguishable particles (see, for example, [4]). The essential difficulty is that, if one mentally labels indistinguishable particles, and one then considers operations on these particles such as swapping their locations, then one has implicitly assumed that the labels persist through time, which contradicts the initial assumption that the particles are indistinguishable. Furthermore, even if one shows that the predictions are invariant under relabelling, one cannot rule out the possibility that the predictions depend on the fact that one has used labels in the first place. Therefore, to guard against inadvertent inconsistencies or implicit assumptions, it is desirable to refrain from such labelling. Such a point of view is in accord with the informational view of physical theory, with its insistence that theoretical distinctions be underpinned as far as possible by experimentally measurable data, a view that has historically been pivotal in the development of physics, leading, for example, to the abolition of absolute time, a mental construct that has no known experimentally-accessible physical counterpart.
In recent years, there has been growing appreciation that an informational viewpoint provides a powerful vantage point for establishing the physical basis and range of validity of the mathematical formalism of quantum theory [23][24][25][26]. Substantial progress has now been made, both in deriving much of the quantum formalism from informational principles [27][28][29][30][31][32][33][34], and in identifying informational principles that account for some of the nonclassical features of quantum physics [35,36]. However, the quantum treatment of identical particles, one of the most important features of quantum theory insofar as its application is concerned, has hitherto not been analyzed from an informational standpoint. Our purpose here is to provide such an analysis.
Here we show that the symmetrization postulate can be derived in full generality within a fully informational approach: if two particles are indistinguishable on the basis of information obtainable through measurements on them, we refrain from labelling the particles. Our key postulate is the operational indistinguishability postulate, which establishes a relation between the theoretical description of two different experiments, positing that the amplitude of a process involving several indistinguishable subsystems (hereafter referred to as particles) is determined by the amplitudes of all possible transitions of these particles when treated as distinguishable. Our informational approach allows us to clearly grasp the essential meaning of the symmetrization postulate, and to steer clear of conceptual pitfalls that commonly afflict treatments of this subject [19,20,37].
We shall carry out our derivation in the framework of Feynmanʼs formulation of quantum theory, and thereby derive the symmetrization postulate in terms of amplitudes rather than states [38]. We shall then show how the better known state formulation of the symmetrization postulate can be derived from the amplitude form of the symmetrization postulate, and indicate the insights to which this leads.
The derivation harnesses the framework and methodology recently used to derive Feynmanʼs formalism of quantum theory [32], in two key ways. First, we make use of the insight that Feynmanʼs rules are best regarded as establishing a formal relationship between different experiments [32,39], an insight that lies at the basis of the operational indistinguishability postulate. Second, we employ the simple yet powerful [32,39] requirement of consistency: if it is possible to compute an amplitude in two different ways, they must agree.

Two indistinguishable particles
Consider a quantum system of two particles, each subject to a measurement at successive times t 1 and t 2 . For example, a simultaneous position measurement of both particles can be implemented by a three-dimensional grid of particle detectors which, at each time, yields two position outcomes, one for each particle. We allow for the possibility that these outcomes may take the same (position) value, and we assume (in this idealized setting) that the detector grid is sufficiently extensive and sensitive that it detects precisely two particles at each time, so that neither of the particles goes undetected. Suppose that the measurement performed at t 1 yields outcomes that we shall arbitrarily label ℓ 1 and ℓ 2 , and, similarly, that the measurement at t 2 yields outcomes m 1 and m 2 .
If it is possible to distinguish between the two particles, then there is some measurement (for example, of mass or charge) that we can, in principle, perform on each particle immediately before t 1 and immediately after t 2 which allows us to conclude either that the particle responsible for outcome ℓ 1 is the same as that responsible for m 1 , and similarly that the other particle goes from ℓ 2 to m 2 , which we shall call the 'direct' transition; or that one particle goes from ℓ 1 to m 2 , and the other from ℓ 2 to m 1 , which we shall call the 'indirect' transition 1 (see figure 1).
If, however, the particles cannot be distinguished from one another, we cannot draw such a conclusion. Even the conclusion that 'the particle that generated outcome ℓ 1 was also responsible for outcome m 1 or m 2 , but we do not know which' is invalid since we cannot verify this statement on account of particle indistinguishability. To nonetheless draw such a conclusion would be tantamount to assuming the separate and continued existence of the particles between observations, which assumption our experience with quantum phenomena warns us against. In particular, in the case of electron diffraction at a double-slit, in the absence of which-way detectors at each slit, we cannot infer from the experimental data that the electron passed through one slit or the other; if we nonetheless insist on drawing this inference, we will discover that our predictions are incorrect-we will be unable to account for the observed diffraction pattern [38,39].
As we have no way of 'looking inside' the indistinguishable-particle process, we must treat it as a black box. As a result, we cannot directly compute the amplitude, α, of the indistinguishable-particle process. Again, electron diffraction is analogous: the transition amplitude of an electron from source to screen in the absence of which-way detectors cannot be computed directly. Instead, it is obtained (via Feynmanʼs amplitude sum rule) from the sum of the amplitudes of the two possible transitions through the slits with the detectors present, which amplitudes can be calculated [39].
In the present case of two particles, then, we seek to relate the amplitude of the indistinguishable-particle transition, α, with the amplitudes, α 12 and α 21 , of the 'direct' and 'indirect' transitions of the two distinguishable particles, where each of these particles agree in all their dynamically-relevant properties with the indistinguishable particles. The amplitudes α 12 and α 21 can be computed by a standard sum-over-histories from the initial outcomes ℓ ℓ , 1 2 to m m , 1 2 in the Feynman formalism. In the case where the particles are subject to position measurements, α 12 would consist of a sum over all continuous paths from ′ ′ r r ( , ) 1 2 to r r ( , ) 1  . Now, we cannot relate α to α 12 and α 21 via Feynmanʼs amplitude sum rule since this rule only applies when different experiments are performed on the same system, whereas α refers to indistinguishable particles, while α 12 and α 21 refer to distinguishable particles. So, in order to derive a connection between α and Left: if the particles are distinguishable, one can experimentally determine which transition actually occurs-the 'direct' one (top), or the 'indirect' one (bottom). Note that if, for example, the measurements are position measurements, the arrows do not indicate or imply a spatial trajectory, but merely indicate that a particle that was found at some location at t 1 was found at some other location at t 2 . Right: if the particles are indistinguishable, one cannot say what happened in the intermediate time on the basis of the information experimentally obtained. 1 The scare quotes ('direct', 'indirect') reflect the fact that the classification of a distinguishable-particle transition as direct or indirect is dependent upon our choice of labelling of the two outcomes at each time, which choice has so far been left arbitrary. For example, at time t 1 , we have not specified how one chooses which outcome to label ℓ 1 and which to label ℓ 2 . One might wish to posit a labelling convention to remove this arbitrariness, but we do not require such a convention for the derivation of the symmetrization postulate in the Feynman framework. We shall require a labelling convention, however, in order to make the transition to the state formulation of the symmetrization postulate (see section 2). α 12 and α 21 , we must first assume that, as in the case of electron diffraction, such a relation exists. Accordingly, we postulate that 12 21 where α lies in the closed unit disc, , in the complex plane, and H is a continuous function, with some domain , to be determined (see figure 2(a)). Similarly, when two particles are subject to three successive measurements, we postulate that the indistinguishable-particle amplitude,  γ ∈ , is given by 11 12 21 22 where γ γ γ , , , 11 12 21 and γ 22 are the amplitudes of the corresponding distinguishable-particle transitions (see figure 2(b)), and G is a function, with some domain  − G ( ) 1 , to be determined. Both of these postulates are special cases of the operational indistinguishability postulate stated above.
Additionally, consider the special case that each particle is known to be bound to a separate experiment, and that the particles do not interact with one another (see figure 3). That is, each particle is confined to its own  21 . We postulate that the amplitude of the indistinguishable-particle process is α α H ( , ) 12 21 , where H is a continuous function to be determined. (b) Measurements are performed at three successive times, yielding the indicated outcomes. On the left are shown the four possible transitions of two distinguishable particles, with respective amplitudes γ γ γ γ , , , 11 12 21 22 , compatible with the observed outcomes. We postulate that the amplitude of the indistinguishable-particle process is γ γ γ γ G ( , , , ) 11 12 21 22 . subexperiment, and the two subexperiments are physically isolated from one other. In this case, the particle originating at ℓ 1 will later be detected at, say, m 1 , while the particle originating at ℓ 2 will be detected at m 2 . As an example, let one of the particles be an electron that is bound to a proton in a lab on the earth, while the other be, similarly, an electron bound to a proton in a lab on the moon. In such a case, each subexperiment is statistically independent of the other. That is, the outcome probabilities observed in one subexperiment are independent of the outcome probabilities of the other subexperiment, or even of whether or not the other subexperiment is performed.
In this case of isolated subexperiments, we assume, as is conventional, that we can compute the transition probabilities of each subexperiment without regard for the other. We refer to this as the isolation condition. In particular, if the amplitude of the transition from ℓ 1 to m 1 is u, then the transition probability is simply u | | 2 . Likewise, if the amplitude of the transition from ℓ 2 to m 2 is v, then the transition probability is v | | 2 . Since the subexperiments are statistically independent, the overall transition probability (initial detection at ℓ 1 and ℓ 2 , final detection at m 1 and 2 . Now, we can also compute this overall transition probability using H. In this special case, α = 0 21 but α = uv 12 . Hence, the overall transition amplitude is simply H uv ( , 0), which yields an overall transition probability of H uv ( , 0) 2 . Consistency requires that this agree with the overall transition probability computed above, so that, for all Similarly, if the outcomes ℓ 1 and m 2 comprise one subexperiment, and ℓ 2 and m 1 another, and these subexperiments are isolated from one another, then, for all

Formulation of the functional equations
We shall now show that the above assumptions determine the precise form of symmetrization postulate for two particles. The general case of N particles, together with additional detail and background terminology, can be found in appendix A.
To determine G and H, we repeatedly use the idea that, if it is possible to compute an amplitude in two different ways using Feynmanʼs rules for single systems [32,40] (see also appendix A.2.1) and the assumptions given above, consistency requires that they be equal. Each such call for consistency yields a functional equation. By considering a few special cases, we are able to formulate a set of functional equations that determine G and H.
First, consider a two-stage experiment in which the intermediate outcomes m 1 and m 2 are both atomic [32] (see appendix A for terminology). We can derive the indistinguishable-particle amplitude, γ, in two different ways (see figure 4). Let α α , 12 21 , respectively, be the amplitudes of the direct and indirect transitions in the first stage, and β β , 12 21 the corresponding amplitudes in the second stage. Using the amplitude product rule, we can directly compute the values γ α β = 11 12 12 , γ α β = 12 12 21 , γ α β = 21 21 12 , and γ α β = 22 21 21 . Then, using the G-function defined above  Figure 3. Isolation condition. In the center, two indistinguishable particles are each subject to measurements at times t 1 and t 2 , yielding the indicated outcomes. Each particle is bound to its own subexperiment, which are isolated from one another. Accordingly, the transition probability can be computed in two different ways. As on the left, one can compute the transition probabilities, u | | 2 and v | | 2 , for each of the particles separately, and multiply these to obtain uv | | 2 . Alternatively, as on the right, one can use the H function to compute the amplitude α α = H Hu v ( , ) ( , 0) 12 21 , which yields transition probability H uv Alternatively, we can directly apply the amplitude product rule to the system of identical particles to obtain , . 12

12 21
Equating these two expressions, we obtain a functional equation, the G-product equation valid for all α α β β ∈ S ( , ), . Next, consider a two-stage experiment in which the outcome of one of the measurements performed at t 2 is a coarse-graining of m 2 and ′ m 2 , which outcome we denote ′ m m ( , ) 2 2 (see figure 5). An example of such a coarse-grained outcome would be a Stern-Gerlach measurement where a detectorʼs field of sensitivity encompasses the fields of sensitivity of two detectors, each of which corresponds to a different atomic outcome (see appendix A.1 for further details). Here, α α , 12 21 are respectively the first-stage amplitudes of the direct and indirect transitions compatible with the indistinguishable-particle process through m 2 ; α α ′ ′ , 12 21 are similarly the amplitudes corresponding to the process through ′ m 2 ; and β β , 12 21 are the second-stage amplitudes of the direct and indirect transitions compatible with the indistinguishableparticle process through both m 2 and ′ m 2 . The indistinguishable-particle amplitude, γ, in this case can be computed in two different ways. First, using the amplitude sum and product rules, we can write Alternatively, one can compute γ directly from equation (2):  Figure 4. Origin of the G-product equation. In the center, two indistinguishable particles are each subject to measurements at times t 1 , t 2 and t 3 , yielding the indicated outcomes. The amplitude for this process can be computed in two different ways. On the left are shown the four possible transitions of two distinguishable particles compatible with the observed outcomes. These transitions can each be expressed as series · ( ) combinations of the indicated transitions, yielding amplitudes α β α β α β α β , , ,  (2), the indistinguishable-particle process amplitude is α β α β α β α β G ( , , , . On the right, the indistinguishable-particle process is first decomposed into two smaller indistinguishable-particle processes in series · ( ), with respective amplitudes, α α H ( , ) 12 21 and β β H ( , ) 12 21 , yielding an overall indistinguishable-particle amplitude α α β β H H ( , ) ( 21 . where we have used the amplitude sum rule to compute the amplitude of the four transitions compatible with the indistinguishable-particle process. Hence, we have the functional equation for which the right-hand side lies in . Lastly, we consider an experiment on two identical particles in which the sequence of measurements performed on the particles is reversed, and the measurements follow one another immediately in time (see figure 6). As indicated in the figure, using the amplitude conjugation rule, the amplitude can be computed in two different ways, which can be equated to give .

Solution of the functional equations
We now show that the functional equations (5), (6) and (7), together with equations (3) and (4), lead to the symmetrization postulate for the two particle case. Using equation (5), the left-hand side of equation (6) can be rewritten to give A nontrivial solution requires that β β H ( , ) 12 21 cannot be zero for all β β , 12 21 . Hence, we obtain the additivity relation Figure 5. In the center, a system of two indistinguishable particles is subject to measurements at three successive times. One of the outcomes of the measurements at t 2 is a coarse-graining of atomic outcomes m 2 and ′ m 2 , denoted ′ m m ( , ) 2 2 . The amplitude of this indistinguishable-particle process can be computed in two different ways: (i) on the right, the process is decomposed into two processes combined in parallel ∨ ( ), and each of these in turn is expressed as two processes combined in series · ( ). The amplitudes of each of the four component processes can then be determined using H in terms of the amplitudes of distinguishable-particle transitions, yielding  21 12 , and γ α α β = + ′ ( ) , 22 21 21 21 are computed using Feynmanʼs sum and product rules as indicated. The amplitude of the indistinguishableparticle sequence is then obtained using the G-function to be γ γ γ γ G ( , , , ) 11 12 21 22 . . We are now in a position to determine H. First, equation (8) implies that Now, equations (3) and (4) , and equation (5) then implies that, for all Since ≠ H (1, 0) 0 from equation (3), we can use equation (10) to rewrite equation (9) as 12 21 12 21 To determine the form of H z ( , 0), we note from equation (8) that, for all  + ∈ z z z z , , 1 (12) and (13) can be written as a pair of functional equations where f (·) is a complex-valued function of a complex argument. We are interested in the continuous solutions of these equations for all  ∈ z z , 1 2 for which  + ∈ z z 1 2 . As shown in appendix B, these solutions are The zero solution is inadmissible since, from its definition, = f (1) 1. Therefore Figure 6. Reversal argument. On the top-right, a system of two indistinguishable particles is subject to measurement M first, and then L immediately afterwards, yielding outcomes m m , 1 2 at time t 1 and ℓ ℓ , 1 2 at t 2 . The amplitude of this process can be computed in two different ways. As shown on the left, one can first take the amplitude, α α H ( , ) 12 21 , of the indistinguishable-particle process with the measurements in their original order, and then take its complex-conjugate to get α α H* ( , ) 12 21 . Alternatively, as shown on the bottom, one can compute the amplitude of the reversed process directly in terms of the distinguishable-particle sequences compatible with it. This yields α α H ( , ) 12 * 21  (3) and (4), we see that = H H | (1, 0)| | (0, 1)|; and we note that equation (7) implies that H (1, 0) and H (0, 1) are both real. Therefore Hence, equation (11) 12 21 12 21 where the sign is the only remaining degree of freedom, corresponding to bosons and fermions. Finally, we note that the above result is based on a particular labelling (ℓ ℓ , 1 2 at t 1 , and so on) of the two outcomes at each time. Now, if the outcome labels at t 1 were swapped, the classification of the transitions into 'direct' and 'indirect' would also be swapped, inducing the transformation α α → 12 21 and α α → 21 12 , so that one would obtain the amplitude α α ± 21 12 instead of α α ± 12 21 . But the transformed amplitude α α ± 21 12 yields the same transition probability as α α ± 12 21 . The same conclusion holds true if the outcome labels at t 2 are swapped. Therefore, the particular choice of labelling at t 1 and t 2 is predictively irrelevant, and so can be chosen arbitrarily without loss of generality. Hence, equation (16) is the final result.
This completes the derivation of the symmetrization postulate for two identical, indistinguishable particles as expressed in Feynmanʼs formulation of quantum theory.
The derivation for N particles is given in appendix A, and follows a similar line of argument. For N identical particles, there are N! distinguishable-particles transitions that must be considered, each corresponding to some permutation, π, in the symmetric group S N . Denoting the amplitude of the transition corresponding to π as α π , we obtain the result ∑ π α α α α = π π π π π σ ∈ H ( , ,..., ) (sgn ( )) , where σ = 0, 1 is the only remaining degree of freedom, whose value does not depend on the number of particles, N.

State formulation of the symmetrization postulate
We shall now briefly discuss how the amplitude formulation of the symmetrization postulate leads to the state formulation. We shall thereby obtain important insight into the interpretation of the latter. For concreteness, we shall restrict attention to two structureless particles moving in one spatial dimension; the generalization to many particles with internal structure moving in d dimensions is straightforward. Consider two structureless, identical particles that are detected at locations ′ x 1 and ′ x 2 at time t 1 , and subsequently at locations x 1 and x 2 at time t 2 . We adopt the convention that, at any given time, the locations are labelled such that the leftmost location is labelled with subscript 1 and the rightmost with subscript 2, so that ′ ⩽ ′ x x a b is the amplitude for the transition of two distinguishable particles where one particle ('particle 1′) goes from ′ x 1 at t 1 to x a at t 2 , and the other ('particle 2′) goes from ′ x 2 at t 1 to x b at t 2 . We note that ψ x x ( , ) ID 1 2 is normalized over the so-called reduced configuration space of points x x ( , ) x x ID 1 2 2 1 2 1 2 whereas ψ x x ( , ) 1 2 is normalized over the full configuration space We can, however, for convenience, extend ψ ID over the full configuration space by defining the function . Mathematically, equation (21) is the standard form in which the symmetrization postulate is usually stated. However, as we have derived it from first principles in an informational experimental framework, we are able to able to make a number of important observations. First, the normalization equations (19) and (20), together with equation (18), imply that the functions ψ x x ( , ) Second, it is commonly asserted that x i in the expression ψ ∼ x x ( , ) ID 1 2 is the position of particle i, and that therefore ψ ∼ x x ( , ) ID 2 1 represents the state when the identical particles have swapped positions. This assertion forms the basis of many textbook arguments for the symmetrization postulate 2 . However, as one can see form the derivation above, x 1 and x 2 are merely positions of two identical particles, so that the subscripts 1 and 2 label two locations, not two particles. Moreover, one can see that the use of ψ ∼ ID , which is defined over the full configuration space, rather than ψ ID , is merely a convenience-the state of two identical particles is better represented by ψ ID as it cannot be subject to the above-mentioned misinterpretation since it is only defined over the reduced configuration space (as ψ x x ( , ) , so that there is no possibility of misinterpreting ψ x x ( , ) ID 2 1 as meaning that the first particle is at x 2 and the second is at x 1 ).
Finally, we see that the symmetrization postulate, equation (21), is perhaps better viewed not so much as a constraint (or selection rule) on the space of states that picks out states that can describe indistinguishable particles, but rather as a rule that allows one to construct valid indistinguishable-particle states from certain distinguishable-particle states. In other words, it is a bridge between the two different state spaces, one belonging to distinguishable particles, the other belonging to indistinguishable particles.

Discussion
We have shown that there exists a direct path from the assumption that identical particles are indistinguishable to the symmetrization postulate. The derivation rests on the following assumptions: (1)the operational indistinguishability postulate, which is formalized via equations (1) and (2)  The present derivation differs substantially from the topological approach, both in the nature of its assumptions and in its conclusions. While the topological approach considers only structureless particles, the present derivation makes no assumptions concerning the nature of the particles involved, so that the result is not restricted to structureless particles but applies to any quantum system, such as a system of identical particles with spin, or a system of identical abstract finite dimensional subsystems. In addition, while the topological approach depends critically on the assumption that particles cannot coincide, the present derivation makes no such assumption. Finally, while the topological approach relies on a generalization of Feynmanʼs sum rule for multiply-connected spaces, the present derivation relies only on the standard quantum formalism.
With respect to their conclusions, the topological approach implies that the symmetrization postulate is valid in three spatial dimensions, but fails in two dimensions, allowing for a one-parameter family of so-called anyonic behaviors. In contrast, the present derivation implies that the validity of the symmetrization postulate is unrelated to the dimension of space, and, in particular, that the symmetrization postulate is valid in two dimensions.
We emphasize that our conclusion (that, contra the topological approach, the symmetrization postulate is valid in two dimensions) is not at odds with the experiment observation of anyonic behaviour in such phenomena as the fractional Hall effect. Although such phenomena are often believed to confirm the idea that the symmetrization postulate breaks down in two dimensions, anyonic behaviour arises generically when particles are confined to two dimensions if the particles are unable to coincide, irrespective of whether the particles are identical or nonidentical [42,43]. Hence, if one is dealing with a system of identical particles that cannot coincide, and these particles are confined to two dimensions, then the anyonic behaviour will make itself felt through the α π . That is, the presence of anyonic behaviour does not preclude the validity of the symmetrization postulate in two dimensions.
To understand this point in more detail, let us briefly consider two structureless, distinguishable particles, with positions r 1 and r 2 , which cannot coincide and are confined to two dimensions. In this case, the relevant configuration space is the so-called relative configuration space of = − R r r 2 1 where the origin, = R 0, is excluded as the particles cannot coincide. Owing to the exclusion of the origin, two paths from point R A to R B in this space cannot always be deformed into one another, and the set of all paths from R A to R B breaks into a countable infinity of so-called homotopy classes, each class indexed by the winding number of the paths it contains. By Schulmanʼs generalized sum rule for multiply-connected spaces [16], using the results of [3], one finds that the amplitude α for the transition from R A to R B consists of an infinite sum of so-called partial amplitudes , is the sum of the amplitudes of all paths in homotopy class n, where n is the winding number of all paths in the class. The single degree of freedom, ϕ, in the above expression, which arises because of the exclusion of particle coincidence, is responsible for the so-called anyonic behavior.
This degree of freedom naturally carries forward to the case where the particles are identical. The transition for two identical particles from R A to R B in the reduced configuration space is given by α α α = ± 12 21 . In this expression, α 12 is the amplitude of the transition of two distinguishable particles from R A to R B , and hence has the form of equation (22). The amplitude α 21 for the transition from R A to −R B similarly involves ϕ [44]. Hence, the identical-particle amplitude α α α = ± 12 21 also involves a single additional degree of freedom, ϕ, which arises from the exclusion of particle coincidence, and it is this degree of freedom which is responsible for anyonic behavior. Hence, experimental evidence for anyonic behaviour does not require renunciation of the symmetrization postulate, but only exclusion of particle coincidence.
More broadly, the present derivation is a natural extension and application of the ideas previously used to derive Feynmanʼs rules of quantum theory. This demonstrates that the symmetrization postulate is comparable in nature and reliability to the core quantum formalism, contrary to what has previously appeared to be the case.
As we have described previously [39], the methodology used to derive Feynmanʼs rules has been previously used to derive [45] the formalism of probability theory starting from Boolean logic, and has recently been applied to yield important insights in other domains [46]. The successful application of this methodology to the symmetrization postulate leads us to anticipate that this methodology may provide a valuable tool in understanding and constructing physical theory.
In this section, we present the derivation of the symmetrization postulate for N indistinguishable subsystems. This derivation closely follows the derivation for two indistinguishable subsystems given in the main text. For reasons of precision and clarity, we explicitly employ the the operational framework and notation used in [32]. For completeness, we first summarize the key features of this framework, and then describe the extensions we need in order to describe composite systems. We then summarize how one can describe Feynman paths and Feynmanʼs amplitude rules in this operational framework. Having laid this foundation, we present the derivation itself.
A.1. Operational framework As explained in [32,47], the measurements and interactions which can be employed in a given experiment must be carefully circumscribed if they are to lead to a well-defined theoretical model. The overarching requirement is that, in the given experiment, the outcome probabilities of any measurement but the first are independent of any interactions with the system prior to the first measurement, a requirement we refer to as experimental closure. Intuitively, the first measurement 'screens off' the prior history of the system, rendering this prior history irrelevant insofar as making predictions about the outcomes of subsequent measurements in the experiment is concerned. The requirement of experimental closure naturally leads to the following definitions and constraints.
An experimental set-up is defined by specifying a source of physical systems, a sequence of measurements to be performed on a system on a run of the experiment, and the interactions with the system which occur between the measurements. In a run of an experiment, a physical system from the source passes through a sequence of measurements L M N , , ,..., which respectively yield outcomes ℓ m n , , ,... at times t t t , , ,... 1 2 3 . These outcomes are summarized in the measurement sequence ℓ m n [ , , ,...]. To avoid notational clutter, the measurements that yield these outcomes, the times at which these outcomes occur, and the nature of the system that is under examination, are all left implicit in the notation, and so must be inferred from the context. In between these measurements, the system may undergo interactions with the environment.
Over many runs of the experiment, the experimenter will observe the frequencies of the various possible measurement sequences, from which one can (using Bayes' rule) estimate the probability associated with each sequence. We define the probability P A ( )associated with sequence A particular outcome of a measurement is either atomic or coarse-grained. An atomic outcome is one that cannot be more finely divided in the sense that the detector whose output corresponds to the outcome cannot be sub-divided into smaller detectors whose outputs correspond to two or more outcomes. We refer to a detector whose firing generates an atomic outcome as an atomic detector. An example of atomic outcomes are the two possible outcomes of a Stern-Gerlach measurement performed on a silver atom. A coarse-grained outcome is one that results from the firing of a coarse-grained detector, namely a detector whose field of sensitivity encompasses those of two or more atomic detectors. To be clear, a coarse-grained outcome does not mean that one has made a fine-grained measurement, thereby obtaining one of N possible outcomes, and one has then applied a many-to-one map yielding one of less than N possible outcomes, a process that discards information and could be regarded as an informational coarse-graining. Rather, a coarse-grained outcome is that due to the firing of a detector that is a physical coarse-graining of two or more detectors.
Abstractly, if measurement ͠ M has an outcome which results from a detector which is a coarse-graining of atomic detectors whose outcomes are labelled 1 and 2 of measurement M, this outcome of ͠ M is labelled (1, 2). This notational convention naturally extends to coarse-graining of more than two atomic detectors. In general, if all of the possible outcomes of a measurement are atomic, we shall call the measurement itself atomic. Otherwise, we say it is a coarse-grained measurement.
It is important that all of the measurements that are employed in an experimental set-up come from the same measurement set, , or are coarsened versions of measurements in this set. This ensures that all the measurements are probing the same aspect of the system. The set is operationally defined as follows. Measurement M forms a measurement pair with L if in both (a) experiment 1, where measurement M is performed on a given system immediately after it has been prepared using L, and (b) experiment 2, where measurement L is used to prepare the given system, and measurement M is performed immediately afterwards, we have experimental closure, namely that the outcome probabilities of the final measurement in both instances are independent of interactions with the system prior to the preparation. The measurement set, , containing M is the set of all measurements that form a measurement pair with M. Interactions that occur in the period of time between measurements are likewise selected from a set, , of possible interactions, which are such that they preserve closure when performed between any pair of measurements from . Experimental closure requires that the first measurement, M, in an experiment is atomic; the remaining measurements must either lie in the same measurement set as M or be coarsened versions of measurements in this set.
A.1.1. Composite systems. The operational framework sketched above is concerned with measurements that probe the entire system. But, in the case of a composite system, it is possible to perform measurements that only probe particular subsystems of the system. For example, consider a system subject to position measurements at successive times. If, at each of these times, two clicks are registered, we say that the system is a composite system of two particles.
More formally, consider an experimental arrangement in which, at each time, one measurement is performed on each of the two subsystems of a composite system. If, at time t 1 , measurements L 1 are performed on one subsystem and L 2 on the other, and simultaneously yield outcomes ℓ 1 and ℓ 2 , respectively, we shall notate the outcome of the two measurements as ℓ ℓ ( : ) 1 2 . If the measurement L 1 and L 2 are performed at time t 1 , and measurements M 1 and M 2 are subsequently performed at t 2 , the resulting measurement sequence can accordingly be written as If it is possible to carry out measurements immediately before t 1 and after t 2 whose outcomes together determine whether the subsystem detected at M 1 originated at L 1 or at L 2 , we say that these subsystems are distinguishable. Otherwise we say they are indistinguishable, in which case the statement that subsystem 1 originated at L 1 and transitioned to, say, M 1 , is not, in general, operationally meaningful.

A.2. Operationalization of Feynman's paths and Feynman's rules
Consider an experimental set-up in which a physical system is subject to successive measurements L M N , , at successive times t t t , , 1 2 3 , with there possibly being interactions with the system in the intervals between those measurements. Here and subsequently, we assume that the measurements and interactions in any such set-up are selected according to the constraints described above. We summarize the outcomes obtained in a given run of the experiment as the sequence ℓ = C m n [ , , ]. This is the operational counterpart to a Feynman 'path'. We now wish to formalize the idea that set-ups are interrelated in particular ways. In [32], we considered three such relationships. First, the above set-up could be viewed as a series concatenation of two experiments, the first in which measurements L and M occur at times t 1 and t 2 , yielding the sequence ℓ = A m [ , ], and the second in which measurements M and N occur at times t 2 and t 3 , yielding = B m n [ , ]. In order to ensure that experimental closure is satisfied in the second experiment, measurement M must be atomic. Formally, we express this concatenation as where · is the series combination operator. More generally, the series operator can be used to concatenate two sequences provided their initial and final measurements are atomic, and the final measurement and outcome of the first sequence is the same as the initial measurement and outcome of the second sequence. Second, one can consider a set-up which is identical to the one above, except that detectors corresponding to outcomes m and ′ m of M have been coarse-grained, so that one obtains the sequence where ∨ is the parallel combination operation. More generally, the parallel operator can combine any two sequences which are identical except for differing in the outcome of a single measurement in the set-up, provided that this measurement is not the initial or final measurement. Third and finally, we can consider a set-up which is identical to the one above, except that the sequence of measurements has been reversed, so that measurements N M , and L occur at times t t , 1 2 and t 3 , respectively, and yield the sequence ℓ = F n m [ , , ]. We express the relationship of this sequence to C as where ← · is the reversal operator.
A.2.1. Feynmanʼs rules in operational form. From the above definitions, it follows that the operators · and ∨ satisfy several symmetry relations, to which we collectively refer to as an operational logic. In [32], it is shown that Feynmanʼs rules are the unique pair-valued representation of this logic consistent with a few additional assumptions. Writing z X ( )for the complex-valued amplitude that represents sequence X , one finds These are Feynmanʼs rules for measurements on single quantum systems. We remark that the amplitude conjugation rule only holds in the special case where the measurements in sequence A immediately follow one another in time.

A.3. Derivation
The derivation for N indistinguishable subsystems closely follows the derivation for two indistinguishable subsystems given in the main text. The key postulates are as follows. Consider an experiment involving > N 1 , is a function of the amplitudes of the transitions of N distinguishable particles compatible with A. There are N! such transitions, each corresponding to some permutation, π, in the symmetric group S N : in a transition corresponding to π, the subsystem that was measured at L j is later found at π M j ( ) , for = j N 1, 2 ,..., . We notate this transition as π π → 1 , where π 1 is the identity permutation here and subsequently. Accordingly, we write where the π i are the N! distinct permutations in S N , and α π is the amplitude of the distinguishable-particle transition π π → 1 . Similarly, for three stages of measurement, with each stage consisting of N measurements, we postulate that the sequence ℓ = C m n [ , , ], where ≡ n n n n ( : : ...: ) , has amplitude performed at t 2 . If a subset of these subsystems is bound to an isolated subexperiment, we shall assume, as is conventional, that we can compute the transition probability for the subexperiment without regard for the subsystems that are not part of the subexperiment. This is the isolation condition.
We shall consider two special cases. In the first, each subsystem is localized to its own subexperiment (see figure 3). As an example, we may have N electrons, each localized to a widely separated hydrogen atom. In particular, suppose that, for = j N 1 ,..., , and some permutation π i , the subexperiment involving measurements L j and π M j ( ) i is isolated. That is, with certainty, the subsystem originating at outcome ℓ j of measurement L j will later be later detected at some outcome, π m j In the second special case, we have two subsystems localized to their own subexperiment, while each of the other subsystems are localized to their own subexperiment (see figure 7). As an example, we may have two electrons bound to a helium atom, while the other − N 2 electrons are each bound to their own hydrogen atom. Let us suppose that the measurements L L , where Γ α β = π π ij i j is the amplitude of the distinguishable-particle transition π π π → → i j 1 , computed using the amplitude product rule. Alternatively, we can write , is obtained. We can compute its amplitude in two ways.
Let . We restrict attention to the special case where β β = π π r ( ) j j . Then, we can directly apply equation (A.6) to get i i j is computed using the amplitude sum and product rules. Using the G-product equation (equation (A.12)), this can be written ,..., ,..., .
(1) Figure 7. Isolation condition (second special case). In the center, three indistinguishable particles are each subject to measurements at times t 1 and t 2 , yielding the indicated outcomes. There are two subexperiments, one involving measurements L L ,  To determine the values of π Q ( )for all possible permutations π ∈ S n , we first show that π = ± Q ( ) 1, and then show that π π = ′ Q Q ( ) ( )whenever π and π′ are both odd or both even. That is, any two permutations, π π″ , , that are connected by a pair of transpositions have the same Q-value. But every even permutation can be written as a product of an even number of transpositions and the identity permutation. Therefore, all even permutations have the same Q-value, Q e . Similarly, every odd permutation can be written as a product of an even number of transpositions and a given odd permutation. Therefore where π sgn ( ) takes the value +1 or −1 according to whether π is even or odd. Insofar as the probability of the transition of the system of N indistinguishable particles is concerned, the overall sign of H and the complex conjugation are irrelevant. More generally, consider a system, , that consists of subsystems  1 and  2 , where  1 consists of N indistinguishable particles of one type (say, electrons). Suppose, first, that  2 only contains particles that can be distinguished from those in  1 . As described above, let measurements L L ,..., N 1 and M M ,..., N 1 be performed on  1 at times t 1 and t 2 . Additionally, let measurements U and V be performed on  2 at t 1 and t 2 , respectively, yielding outcomes u and v. Let α π be the amplitude of the transition of  from ℓ u ( : )to m v ( : )in which the particles in  1 are treated as distinguishable and make the transition described by π. Then, by the same argument as described above, the amplitude of the process from ℓ u ( : )to m v ( : )where the particles in  1 are treated as indistinguishable is given by α α α π π π H (˜,˜,...,˜) N 1 2 ! , and the transition probability is again unaffected by the overall sign or complex conjugation of H. In the case that  2 does contain, say, M particles that are indistinguishable from those in  1 , the boundaries of  1 must be redrawn to encompass them. The resulting situation, namely a system composed of subsystem ′ 1 , containing + N M indistinguishable particles, and subsystem ′ 2 , containing only particles that are distinguishable from those in ′ 1 , is of the same type the as one previously considered.
Therefore, in general, the overall sign of H and the complex conjugation are irrelevant insofar as predictions are concerned, and can be discarded without any loss of generality. Hence where σ = 0 or σ = 1 is the only remaining degree of freedom, corresponding respectively to bosons and fermions. The above result for the amplitude holds for a particular labelling of the outcomes at times t 1 and t 2 . However, the corresponding transition probability is invariant under relabelling of these outcomes. To see this, suppose that the outcomes at t 1 and t 2 are relabelled such that ℓ ℓ → τ′ {1, 2 ,..., },