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Eur. J. Phys. 19 (1998) 237–243. Printed in the UK PII: S0143-0807(98)87270-X Limitations on the superposition principle: superselection rules in non-relativistic quantum mechanics C Cisneros†, R P Martı́nez-y-Romero‡, H N Núñez-Yépez§k, and A L Salas-Brito¶ † Laboratorio de Cuernavaca, Instituto de Fı́sica, Universidad Nacional Autónoma de México, Cuernavaca Mor, Mexico ‡ Departamento Fı́sica, Facultad de Ciencias, Universidad Nacional Autónoma de México, Apartado Postal 21-726, Coyoacán 04000 DF, Mexico § Instituto de Fı́sica ‘Luis Rivera Terrazas’, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, CP 72570, Puebla Pue, Mexico Received 12 September 1997, in final form 16 February 1998 Abstract. The superposition principle is a very basic ingredient of quantum theory. What may come as a surprise to many students, and even to some practitioners of the quantum craft, is that superposition has limitations imposed by certain requirements of the theory. The discussion of such limitations arising from the so-called superselection rules is the main purpose of this paper. Some of their principal consequences are also discussed. The univalence, mass and particle number superselection rules of non-relativistic quantum mechanics are also derived using rather simple methods. 1. Introduction Linearity is one of the most basic properties of quantum theory: any set Pof states, |αi, of a quantum system may be linearly superposed to obtain a new state |Ai = α aα |αi, where the aα are complex numbers, the probability amplitudes for the system to be found in the state |αi. Linear combinations like |Ai are usually referred to as coherent superpositions or pure states (Sakurai 1985). A brief discussion of coherent as contrasted to incoherent superpositions or mixtures, is given in section 2. However, despite being fundamental, it is easy to realize that the superposition principle cannot hold unrestrictedly in every situation. For example, think of photons and electrons: then, no matter what we have just said about the superposition principle, no one has ever made sense of any superposition of electronic with photonic states (Sudbery 1986, SalasBrito 1990). How can we understand this peculiarity within the general structure of quantum theory? The justification for such fact was found long ago (Wick et al 1952) and involves k On sabbatical leave from: Departamento de Fı́sica, UAM-Iztapalapa, Mexico; e-mail: [email protected] . ¶ Corresponding author. On sabbatical leave from: Lab. de Sistemas Dinámicos, Departamento de Ciencias Básicas, UAM-Azcapotzalco, Mexico; e-mail: [email protected] or [email protected] . 0143-0807/98/030237+07$19.50 © 1998 IOP Publishing Ltd 237 238 C Cisneros et al the curious phenomenon referred to as a superselection rule. To give a brief description, a superselection rule exists whenever there are limitations to the superposition principle (Kaempfer 1965, Sudbery 1986), that is, whenever certain superpositions cannot be physically realizable. Moreover, the very existence of superselection rules explains why we can treat certain observables like the mass of a particle as parameters rather than as full-fledged operators in non-relativistic quantum mechanics (NRQ). It is somewhat peculiar that despite the additional insight into the subtleties of quantum theory that might be offered by superselection rules and of the fundamental role they have played in explaining puzzling phenomena in various areas of physics (Streater and Wightman 1964, Pfeifer 1980, 1983, Putterman 1983, Salas-Brito et al 1990), they have not yet found a place in many of the excellent NRQ textbooks available. In this paper we want to contribute to alleviate the situation by first explaining how superselection rules fit in the general structure of quantum mechanics and then by offering some examples in the context of NRQ. 2. Superselection rules in quantum theory 2.1. Superposing photons and electrons is forbidden Let us begin by analysing the possibility of coherent superpositions of photon and electron states. Any such superposition could, at least in principle, be written in the form (1) |9i = ap |photoni + ae |electroni. Recall that whereas electrons have spin 21 , which makes them fermions, photons have spin 1, making them bosons instead. Bosons and fermions differ in many ways, but, for our purposes, the crucial difference is in their behaviour under rotation. Consider a bosonic state and rotate the system by 2π, a rotation which is expected to leave the physical content of this—and any other—state unchanged. Under this rotation the bosonic state does indeed not change; but a fermionic state ends multiplied by −1 under that same rotation (Feynman et al 1965, Sakurai 1985). Even so, there is nothing to worry about since the quantum states are determined up to a normalizing factor and thus the final state is also physically indistinguishable from the initial one. However, if rather than rotating a fermionic or a bosonic state separately, we rotate the whole superposition |9i by 2π then the rotated state becomes R(2π)|9i = |9 0 i = ap |photoni − ae |electroni. (2) Given that the probabilities associated with the states (1) and (2) differ, that is, as h9|9i 6= h9 0 |9 0 i, the superposition becomes essentially different from the original state after the 2π rotation. This implies that a superposition of the form (1) is devoid of physical meaning unless ap = 0 or ae = 0, or, in other words, only if coherent superpositions are not allowed. Thus, to be consistent with the properties of rotations, quantum theory is forced to restrict the superposition principle, i.e. to impose a superposition rule between photons and electrons or, to state the restriction in more general terms, to forbid superpositions of any fermionic with any bosonic states whatsoever. This superselection rule is usually called univalence (Wick et al 1952, Streater and Wightman 1964). To avoid possible misunderstanding, it is important to indicate that there is a way of making sense of a state like (1), not as a coherent superposition but as one of the mixed states or incoherent superpositions. Equation (1) can have meaning as a sort of density operator (see subsection 2.2) describing a statistical ensemble with a fractional population ap of photons and a fractional population ae of electrons (Landau and Lifshitz 1977). For example, this can be necessary to describe the situation within a switched-on floodlight. 2.2. Coherent versus incoherent superpositions Before proceeding any further, it is convenient to review the difference usually made in quantum mechanics between coherent and incoherent superpositions. Let us take as example the case of Limitations on the superposition principle 239 spin- 21 particles; the spin of these can be found pointing up |+i or down |−i along any direction; the most general spin state can be expressed as |φi = a+ |+i + a− |−i, where the numbers a+ and a− are assumed to comply with |a+ |2 + |a− |2 = 1. This superposition describes a state with a well-defined spin. The value of any observable O in the pure state |φi is the quantum expectation value ∗ ∗ hφ|O|φi = |a+ |2 h+|O|+i + |a− |2 h−|O|−i + a+ a− h+|O|−i + a− a+ h−|O|+i. (3) The last two terms in this equation are the interference terms; they are an unavoidable characteristic of pure states. However, pure states are not general enough to describe every possible situation; sometimes the need to describe a mixed state might arise. For example, if the spin of some unfiltered beam of particles is required, a mixture is necessary to describe the random spin orientations in the beam—note that such random behaviour has nothing to do with quantum features. To grasp the situation better, let us think of what we should expect of a spin measurement in an unfiltered beam? It is clear that any measurement should produce the classical (i.e. non-quantum) average value hSi = + 21 h̄w+ − 21 h̄w− = w+ h+|S|+i + w− h−|S|−i (4) where S is the spin operator, and the numbers w+ and w− respectively stand for the ratio of particles with spin up (+ 21 h̄) or down (− 21 h̄) in the beam. This is as we should expect since the numbers w+ and w− are just statistical weights describing the population in the beam. This expression may suggest that, in dealing with mixtures, it is convenient to introduce the density operator (Landau and Lifshitz 1977) ρ ≡ w+ |+ih+| + w− |−ih−| (5) to characterize the state; normalizing (5) means that w+ + w− = Tr(ρ) = 1. With such characterization, the measurement of any property O is calculated using not the quantum mechanical expectation value as in pure states, but by taking the trace of the product of O times ρ: hOi = Tr(Oρ) = w+ h+|O|+i + w− h−|O|−i. (6) This is the ensemble average of O in the mixture (4). The important point to note is that this expression lacks the interference terms of the quantum expectation value (3); we can say that in mixtures the relative phase of the individual states does not matter. As the next section shows, in systems with superselection rules there necessarily are states which can be interpreted only as incoherent superpositions, and, therefore, which can never interfere with one another. 2.3. Superposition rules in the formalism of quantum theory If in a quantum system there exist a hermitian operator G which commutes with every observable O in the system (i.e. it conmutes with the basic operators p, q, and possibly the spin), and if G is found not to be a multiple of the identity, then, it is said that a superselection rule induced by G operates in the system. This is in fact closely related to the failure of the superposition principle (Streater and Wightman 1964, Lipkin 1973, Sudbery 1986) as the next subsection shows. Since G commutes with every observable, it commutes, in particular, with the Hamiltonian and it necessarily corresponds to a conserved quantity. We must distinguish this property of G from the corresponding one of ordinary conserved quantities like the energy, since physically realizable states exist that are not eigenstates of the Hamiltonian; when a superselection rule induced by a operator G acts on a system every physically realizable state is an eigenstate of G and not every possible superposition can represent a physical state. The property of commuting with every observable operator makes the variable G sharply measurable in every situation, thus being rather similar to a c-number; known operators of this sort in NRQ are, 240 C Cisneros et al for example, the mass, the electric charge, the particle number and the chirality of a molecule (Kaempfer 1965, Lipkin 1973, Pfeifer 1980, Primas 1981). Let us denote the eigenstates and eigenvalues of G, as |gm ; αm i and gm , respectively; αm stands for every other quantum number needed to specify the states. As every operator commutes with G its eigenstates can always be chosen as eigenstates of the rest of observables of the system; thus, any state |Si can always be expressed as a superposition of the eigenstates of G: X am |gm ; αm i. (7) |Si = m The Hilbert space of the system, H, is naturally decomposed into a collection of disjoint and mutually orthogonal subspaces Hg (since they correspond to different quantum numbers of a Hermitian operator, and thus hgs ; αs |gm ; αm i = 0), each identified by a different eigenvalue of G. The Hg subspaces are called the coherent subspaces or sectors induced by the superselection rule. It is now easy to see that the direct sum (Shilov 1977) of all the coherent subspaces H = H1 ⊕ H2 ⊕ · · · ⊕ HN comprises the whole Hilbert space of the system. One must be somewhat cautious with the statements like (7), because, although the expansions are allowed in principle, the very existence of the superselection rule forbids any meaningful superposition of states in different coherent subspaces. That is, a superselection rule forbids superposing states with different values of the eigenvalue gm , as shown in subsection 2.4. An additional consequence of the existence of a superselection rule is that not every Hermitian operator may represent an observable. To be an observable the operator should not cause transitions between the different coherent sectors, as we argue in subsection 2.5. 2.4. Impossibility of superposing states belonging to different coherent sectors To show that any superposition of eigenstates of G with different eigenvalues is not physically P admissible (Salas-Brito 1990), consider first that, on the contrary, the state |ui = m um |gm ; αm i is physically sound; however, since G commutes with every other observable, |ui should be an eigenstate of every operator in a complete set of commuting observables of the system. Denote this set as Os , s = 1, . . . , N; thus Os |ui = os |ui. As a consequence of the superselection rule, the state |gm ; αm i is also an eigenstate of G with eigenvalue osm ; then, another way of express the action of Os on |ui is X Os |ui = um osm |gm ; αm i. (8) m However, all the |gm ; αm i being linearly independent of one another, P the previous equations imply that osm = os for every m value. Next consider the state |u0 i = m um exp(iδm )|gm ; αm i, which is just |ui with the phases of its components modified, and apply the operator Os to it; we easily obtain X um exp(iδm )|gm ; αm i = os |u0 i. (9) Os |u0 i = os m On comparing the previous expressions, we can see that both states, |ui and |u0 i, have precisely the same quantum numbers; i.e. these two states are completely indistinguishable from one another. But they are also completely different vectors in Hilbert space! The only way of avoiding a contradiction is by altogether forbidding superposition of states in different coherent subspaces. This proves the statement and establishes the connection between the existence of G-operators and limitations on the superposition principle. Note also that the only way of give content to superpositions like |ui or |u0 i is by interpreting them as mixtures, never as pure states. Thus, in a system with superselection rules, the superposition principle only holds within each coherent subspace; superpositions from different coherent subspaces can never describe physical situations. Limitations on the superposition principle 241 2.5. Non-existence of transitions between different sectors It is very easy to show the vanishing of any matrix element of an observable O between states in different coherent sectors Hn and Hm , m 6= n. To accomplish this, keep in mind that that the eigestates of G can have common eigenvalues with any operator in the system, thus hgm ; αm |O|gn ; αn i = ohgm ; αm |gn ; αn i. (10) However, states with different quantum numbers are orthogonal hgm ; αm |gn ; αn i = 0. Thus no observable can ever induce transitions between states in different coherent sectors of the Hilbert space. This means, for example, that according to the univalence superselection rule, it is impossible to produce a single fermion from a single boson in any conceivable NRQ process. 2.6. The most general superselection rules Superselection rules are very important properties of quantum systems. The most general superselection rules that, it is believed, must hold in any quantum theory are associated first with the existence of fermions and bosons, i.e. univalence, and then to every absolutely conserved quantum number: there are superselection rules associated with the electric charge, the baryon number, and the three lepton numbers (Lipkin 1973, Sudbery 1986). This means that all physical states of the basic building blocks of the universe should be sharp eigenstates of the operators associated with those properties. But even within the realm of NRQ there are very interesting superselection rules, which can be obtained by such simple calculations that they could be part of almost any intermediate NRQ course. This is demonstrated explicitly in the next section. 3. Superselection rules in non-relativistic quantum mechanics The superselection rules acting in non-relativistic quantum mechanics and the operators G associated with them are discussed in this section. We briefly discuss first the univalence superselection rule separating the bosonic from the fermionic sectors in the Hilbert space of NRQ, and then the superselection rule separating the different mass sectors, which allow the mass to be treated as just a parameter in the Schrödinger equation. The electric charge superselection rule also operates in NRQ but, as its derivation involves field operators (Aharonov and Susskind 1967, Strocchi and Wightman 1974), the derivation of the rule is beyond the scope of this paper. 3.1. Univalence superselection rule In subsection 2.2, following the arguments of Wick et al (1952), we have already sketched a proof of univalence. It is possible to rederive this superselection rule using arguments closer to the spirit of NRQ. This has been excellently done by Müller-Herold (1985), so here we only want to point out a few things that do not follow directly from the discussion in subsection 2.2. Let us mention that univalence in NRQ can be regarded as a direct consequence of the existence of indistinguishable particles and that the operator associated with it is the permutation operator P interchanging particles in any multiple particle state (Lipkin 1973). The whole Hilbert space of non-relativistic quantum mechanics thus splits as the direct sum of fermionic and bosonic coherent subspaces, i.e. as the direct sum of the eigenspaces of P . 3.2. Superselection rule separating different mass values In dealing with the Schrödinger equation, mass is mostly assumed to be a c-number (mass is sometimes assumed to be an operator, for example, it is convenient to do so for describing the propagation of electrons in a crystal (Ashcroft and Mermin 1976)); moreover, states 242 C Cisneros et al superposing different masses are conspicuously absent in NRQ. Mass seems to comply with a superselection rule; this follows from the Galilean invariance of NRQ as originally noted by Bargman (1954). The change in a state function when both a displacement of the origin by the constant quantity r and a change between two reference frames moving with respect to one another at a constant velocity v occur simultaneously (that is, when the Galilean transformation q 0 = q +v t+r is performed) can be represented by the unitary operator UG = exp(imv·q 0 −imv 2 t/2) (Kaempfer 1965, Landau and Lifshitz 1977). The action of UG on a quantum state |9i is (11) |9 0 i = UG |9i = exp im(v · q 0 − v 2 /2) |9i. Therefore h9 0 |9 0 i = h9|9i and Galilean invariance is manifest in non-relativistic quantum mechanics. Selecting r = 0, specializes (11) to the operator associated with a pure Galilean transformation, Uv ; whereas selecting v = 0, specializes it to the operator associated with a pure spatial displacement, Ur . It must be clear that the combined action on |9i of a spatial displacement Ur , a pure Galilean transformation Uv , the opposite displacement U−r and the inverse Galilean transformation U−v , just amounts to an identity transformation. Its net effect is again just a change in the phase of the state: UI |9i = exp(imv · r )|9i, where we have defined UI ≡ U−v U−r Uv Ur . The transformation UI is seen to be of trivial consequences as long as we are dealing with definite mass states; the change becomes crucial, however, if we consider states |φ1 i and |φ2 i corresponding to different masses m1 and m2 . Since any superposition |φi = a1 |φ1 i + a2 |φ2 i transforms under the identity-equivalent transformation UI as |φ 0 i ≡ UI |φi = a1 exp(im1 r · v )|m1 i + a2 exp(im2 r · v )|m2 i (12) 0 you can see that if m1 6= m2 the states |φi and |φ i become essentially different. We have to conclude that mass induces a superselection rule in NRQ and, therefore, that mass can always be sharply measured in any situation. Thus NRQ is utterly inadequate for describing states endowed with a mass spectrum or for describing transformations between particles. 3.3. Particle number superselection rule Given the mass superselection rule, it is very easy to obtain the closely related superselection rule separating states with different particle numbers in Fock space (Lipkin 1973). The proof follows from the following argument (Müller-Herold 1985), given a state |ni describing n identical particles, each with mass m, this state should transform, under the Galilean transformation U (I ) of subsection 2.2, as U (I )|ni = exp(inmv · r )|ni. From the fact that the states of n particles have a total mass nm different, in general, from the mass n0 m of n0 particle states, the argument of the previous section unavoidably leads to the conclusion that any superposition of n-particle with n0 -particle (n 6= n0 ) states must be a mixed state. The G-operator associated with this superselection rule is the particle number operator N . 4. Summary Superselection rules have been introduced and its principal consequences have been discussed within the framework of non-relativistic quantum mechanics. We have exhibited how certain quantum requirements are capable of constraining the applicability of the superposition principle. Using the basic machinery of quantum mechanics we have derived three of the basic superselection rules that operate in that theory, namely, univalence, mass and particle number. It is surprising that these very interesting and important ideas are almost never discussed in an intermediate course in quantum mechanics. Our explanation for this curious situation is that, originally, superselection rules were introduced in the context of quantum field theory and they were considered features of infinite dimensional theories. On the other hand, perhaps some theoretical prejudices made people believe that it was impossible to find Limitations on the superposition principle 243 superselection rules in simple quantum systems. However, we now know that one can find them even in rather simple systems; for example, in the one-dimensional hydrogen atom with interaction V (x) = −e2 /|x| where e is the electronic charge (Núñez-Yépez et al 1988, 1989, 1987, Martı́nez-y-Romero et al 1989). Acknowledgments This work has been partially supported by CONACyT (grant 1343P-E9607). ALSB wants to thank J L Carrillo-Estrada and L Fuchs-Gómez for their one-week hospitality at, respectively, the Instituto de Fı́sica and the Facultad de Ciencias Fı́sicas y Matemáticas of the BUAP where this work was partially written. We are also extremely grateful to P Pfeifer for providing us with a copy of his dissertation. The useful comments of P A Terek, G Billi, K Quiti and F C Bonito are gratefully acknowledged. 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