On the nucleon paradigm: the nucleons are closer to reality than the protons and neutrons
aa r X i v : . [ phy s i c s . g e n - ph ] J un On the nucleon paradigm:the nucleons are closer to realitythan the protons and neutrons
Fedor Herbut
Serbian Academy of Sciences and Arts, Serbia, Belgrade, Knez Mihajlova 35
Abstract
There is a widespread delusion that in theoretical nuclear physics protonsand neutrons are the real thing, and nucleons are not more than a mathe-matically equivalent formality. It is shown that, on the contrary, nucleonsare the real thing, because only a part of the theory is essentially identicalto proton-and-neutron theory, whereas the remaining part is physically rele-vant. The approach is general. Thus, this is a paradigm of relation of a widerand a more narrow theory, so that the wider theory describes reality better.Also the relation of disjoint domains to the exclusion principle is clarified.A general fermion theory of how to distinguish identical particles is presented.
Keywords:
Reality of physical theory; distinguishing identical particles; exclusion principle and spatialseparation
Nuclei with electron shells make up atoms and molecules, and further all theobjects of the world of classical physics that we are familiar with.What do nuclei consist of, are they protons and neutrons, or nucleons? Ibelieve that many physicists would readily opt for the former. Some wouldchoose the latter.Protons and neutrons differ in mass, electric charge, and in gyro-magneticcoefficient. It is hard to believe in the reality of one, apparently fictitiousparticle, the nucleon, of which the proton and the neutron are two izospin-
E-mail address: [email protected] π = 3 . ... is a good paradigmatic notion. One knows how to improve onthe approximation, but not how to write down the precise value.Two points follow from the relation of ontology and epistemology. Firstly,a theoretical description appears also in versions (b) and (c). In (b) it is allthat there is in quantum physics (a rather poor ontology); in (c) one couldn’tcare less about ontology.Secondly, one must clarify how one expects to ’improve’ one’s descriptionof reality.My answer is that a ’better’ theory must be wider , and, in some sense,it must ’contain’ the former theory, and it must give ’more’ . The lattermust be physically meaningful .In this article ”the nucleon paradigm” (from the title) is conceived as a2heory of nucleons in which it is precisely defined how it is wider and howit contains the description of protons and neutrons, and it is shown what itgives ’more’ then a proton-neutron theory in a physically meaningful way.Actually, this article has a greater ambition. It will give answer to an-other important question that concerns the exclusion principle of Pauli. Onewonders whether one should, perhaps,(i) anti-symmetrize all nucleons in the universe since they are all identicalparticles of the same kind;(ii) or, one should anti-symmetrize only the nucleons that are as close asthose in one and the same nucleus.In case of (i), one must be able to show that though one anti-symmetrizesall nucleons in the universe, when relevant spatial domains are considered,it boils down to anti-symmetrizing special clusters of nucleons separately.In this way then, version (i) would ’contain’ version (ii). The ’more’ wouldcome from the fact that the Pauli principle is based only on the identity ofthe particles and their spin without intrusion of spatial concepts as in (ii).The theoretical framework that is going to be presented covers both cases(i) for the nucleons and case (i) for their anti-symmetrization. It will be po-tentially valid for any kind of identical particles, fermions or bosons.The exposition of the ’wider’, and hence closer to reality, theory in bothmentioned cases will draw on a general theory of ’distinguishing identicalfermions’ explained in Appendices A and B.The nucleon paradigm of two nucleons was discussed in previous work(Herbut 2001). If the reader finds the jump to the general case in this arti-cle, particularly Appendix B, to steep, he (or she) is well advised to reed thementioned previous article first. But it treats only a very special case of thepresent theory. Let a few remarks give a short historical outline of thinking that has led tothe nucleon paradigm.Following Jauch (1966), one can distinguish intrinsic and extrinsic roperties of particles. According to him, identical are those particlesthat have equal intrinsic properties .Jauch’s criterion seems to suggest that, to obtain a wider theory, oneshould treat some intrinsic properties of the particle as extrinsic. But thiswould be in vain unless it had a surplus of physical meaning.De Muynck (1975) remarks on Jauch’s criterion, ”an intrinsic propertymay show up dynamical behavior”, and turn out to be extrinsic like theproton and neutron states.It all depends on the experimental conditions. Mirman (1973) made theimportant claim that distinguishability of identical particles is essentially anexperimental notion.Let us resort to quantum-mechanical state spaces (complex Hilbert spaces).They are the natural mathematical framework for quantum state vectors (el-ements of the space of unit norm) or, more popularly, wave functions (statevectors in coordinate representation). They are quantum pure states (asopposed to mixed states).The single-nucleon state space H nu has three tensor-factor spaces :the orbital (or spatial) one H orb , which is a countably infinitely dimensionalcomplex or a separable Hilbert space , the two-dimensional spin one H sp ,and the two-dimensional isospin one H iso : H nu ≡ H orb ⊗ H sp ⊗ H iso . (1 a )This is so because the interactions between nucleons do not depend on theorientation of the z-component of spin, and, formally, the same is valid forthe orientation of the third component t of the isospin (so-called chargeindependence of the strong interaction).Intuitively speaking, two protons, two neutrons or a proton and a neutron,act equally as far as so-called strong interaction is concerned. (This is notso for electromagnetic and weak interaction.) The proton is the counterpartof spin-up, i. e., it has t = 1 / h ), whereas the neutronhas t = − / proton and the neutron : H pr ≡ H orb ⊗ H sp , (1 b ) H ne ≡ H orb ⊗ H sp . (1 c )4he difference in the three quantum-mechanical state spaces (1a), (1b) an(1c) is in the intrinsic properties of the particles, i. e., properties that donot enter the quantum-mechanical formalism, but underly it .The nucleon is intrinsically only a barion (barion quantum number +1 ).Mass, electric charge and gyromagnetic factor are extrinsic properties. Theyenter the formalism; they depend on the state of the particle.On the other hand, the proton has intrinsically the well-known mass,electric charge and gyromagnetic factor; whereas the neutron has, also in-trinsically, its corresponding quantities.It is customary in nuclear physics to denote the number of protons in anucleus by Z, that of the neutrons in it by N, and their sum by A. To beconsistent with our general notation in Appendix B, we write N pr , N ne and N nu instead of Z, N and A respectively.The N nu -identical-fermion state space is H nu ...N nu ≡ A ...N nu ⊗ ,N nu Y n =1 H nun , (2)where A ...N nu denotes the anti-symmetrizer over all N nu particles (cf(A.2) in Appendix A). One should note that the index n refers to the n -thparticle in the (formal distinct-particle) N nu -nucleon state space Q ⊗ ,N nu n =1 H nun .The superscript reminds of the intrinsic properties that define the identicalparticles.One should note that the state space H nu ...N nu applies to the entire so-called isobar family of nuclei, i. e., to all those nuclei that have the givennumber N nu of nucleons. It begins with N pr = N nu , N ne = 0 , andit ends with N pr = 0 , N ne = N nu . Besides these extreme cases, alsomany other members of the isobaric family usually do not exist in nature.(If obtained artificially, they, being unstable, decay with a certain half life.)On the other hand, the proton-neutron description takes place in a one-nucleus proton-neutron state space H pr,ne ...N nu ≡ H pr ...N pr ⊗ H ne ...N ne , (3 a )where H pr ...N pr ≡ A ...N pr ⊗ ,N pr Y n =1 H prn , (3 b )5 ne ...N ne ≡ A ...N ne ⊗ ,N ne Y n =1 H nen . (3 c )Here A ...N pr and A ...N ne denote the anti-symmetrizers, which are nowapplied separately to the two kinds of particles. (This is usually called the’exclusion or Pauli principle’, whereas the above nucleon anti-symmetrizer in(2) is called the ’extended exclusion’ or ’Pauli principle’.) Finally, both theproton and the neutron have the same single orbital-spin space (cf (1b) and(1c)).To see how the N nu -identical-fermion description in H nu ...N nu can be’restricted’ to the proton-neutron description in H pr,ne ...N , or how the former ’contains’ the latter (cf the Itroduction), distinguishing projectors arerequired (cf Appendix B).The single-nucleon distinguishing projectors in H nu (cf (1a)) arethe eigen-projectors Q + nu and Q − nu of t , (the third-projection of isospinby analogy with s z of spin). The spectral decomposition of t is t = (1 / Q + nu + ( − / Q − nu . (4)Actually, all these operators act in the tensor-factor space H iso (cf (1a)),and they are multiplied tensorically by the identity operators in the orbitaland in the spin factor spaces to be determined in the entire single-nucleonstate space H nu . One uses the same notation in H iso and in H nu .(One can see from the context in which space they act.)To obtain the distinguishing projectors, with the physical meaning ofthe distinguishing property , in the many-nucleon and the many-proton-neutron spaces, for a fixed nucleus , i. e., with fixed numbers N pr and N ne , the procedure goes as follows (cf Appendix B).First we define the distinguishing many-particle projector in the N nu -nucleon space Q ⊗ ,N nu n =1 H nun (cf (1a)). Q ...N nu ≡ (cid:16) ⊗ ,N pr Y n =1 Q + n (cid:17) ⊗ (cid:16) ⊗ ,N nu Y n = N pr +1 Q − n (cid:17) . (5)In the extreme cases of no neutron or no proton, the formula, of course,requires slight obvious changes. 6ne should note that Q ...N nu Q ⊗ ,Nun =1 H nun is not a subspace of the iso-bar family state space A ...N nu Q ⊗ ,Nun =1 H Nun . It is a subspace of the for-mal distinct-particle space Q ⊗ ,Nun =1 H Nun , and it is isomorphic with theone-nucleus proton-neutron state space H pr ...N pr ⊗ H ne ...N ne (cf (3a)). Theisomorphism at issue is obtained by the transition from the one-nucleonspace Q + H nu (cf (1a)) to the proton state space H pr (cf (1b)) and from Q − H nu to the neutron state space H ne (cf (1c)). This amounts to convert-ing the relevant extrinsic properties into intrinsic ones (cf Jauch 1966).In the isobar family state space H ...N nu = A ...N u Q ⊗ ,N nu n =1 H nun onehas the following symmetrized distinguishing projector (a sum of orthogonalprojectors) that is the counterpart of Q ...N given by (5): Q sym ...N nu ≡ ( N pr ! N ne !) − (cid:16) X p ∈ S Nnu P ...N nu Q ...N nu ( P ...N nu ) − (cid:17) , (6)where by S N is denoted, as customary, the symmetric group (or group ofpermutations) on N objects.The next step is to determine the subspace of the N nu -identical-fermionstate space H Nu ...N Nu (cf (2)) that is isomorphic with the proton-neutronspace of a fixed nucleus . It is (cf Theorem 1 and relation (B.5)): H id ...N nu ≡ Q sym ...N nu A ...N nu ⊗ ,N nu Y n =1 H n . (7)As it was stated, the proton is the nucleon with t = 1 / t = − / Q + H nu and Q − H nu are the proton and neutron state spaces respectivelyas subspaces of H nu (cf (1a)). To be more specific, Q + H nu is isomorphicwith the proton space H pr and Q − H nu with the neutron one H ne (cf(1a-c)). In practice, this isomorphism means, for both particles, to omit theisospin tensor factor space in (1a), and take over the orbital and spin spacesunchanged. Particularly, all orbital-spin operators remain unchanged.The one-nucleus state space in the proton-neutron description (3a),with the intrinsic proton-or-neutron properties, is not a subspace of the N nu -nucleon distinct-particle state space Q ⊗ ,N nu n =1 H n . But, if one rewrites(3a) in its isomorphic extrinsic form , based on the insight of the precedingpassage, as A ...N pr A ( N pr +1) ...N nu Q ...N nu ⊗ ,N nu Y n =1 H nun (8)7cf (5)), then it is.On ground of the general Theorem 1 in Appendix B, the one-nucleusstate space H id ...N nu given by (7), which is a subspace of the all-nucleonspace referring to the isobar family, and the one-nucleus state space (8) are isomorphic , and the isomorphism acts as follows : h(cid:16) N nu ! . N pr ! N ne ! (cid:17) / Q ...N nu i H id ...N nu = A ...N pr A ( N pr +1) ...N nu Q ...N nu ⊗ ,N nu Y n =1 H nun . (9)When weak interaction does not play a role , i. e., when no β -radioactivity is taking place, then the distinguishing property Q sym ...N ispossessed by any quantum state of the nucleus. Namely, this property phys-ically simply says that there are N p protons and N n neutrons in the N nu -nucleonic nuclear state ( N nu = N p + N n ). Hence, one can transfer thequantum-mechanical description from the first-principle completely antisym-metric all-nucleon state space given by (2) (in which the so-called ’extendedPauli principle’ is valid) to the effective distinct-cluster space given by (8),or even to the further isomorphic state space H pr,ne ...N (cf (3a)). We have twoclusters here, to utilize the terminology of the general theory in Appendix B,that of protons and that of neutrons.When weak interaction (or β -radioactivity) has to be taken into account,the single-particle spaces H pr and H ne (relations (1b) and (1c) respec-tively) have to be replaced by the doubly dimensional nucleon space H nu given by (1a).So-called weak interaction turns a neutron into a proton or vice versa within a nucleus. One observes this as − β or + β radioactivity (emission ofan electron with a neutrino or emission of a positron with a correspondingneutrino) respectively. This displaces the nucleus in question to a neighboringone with one proton more and one neutron less or vice versa . The point tonote is that this takes place within a barion family of nuclei with a fixednumber of nucleons, i. e. within the state space A ...N Q ⊗ ,N nu n =1 H nun (cf (2)).Mathematically, as known from textbooks on quantum mechanics describ-ing spin, the operators t + ≡ t + it and t − ≡ t − it , the counterparts8f s + ≡ s x + is y and s − ≡ s x − is y , map a neutron state into a protonone (with the same spatial and spin sate) and vice versa (cf Preston 1962).The transformation of a proton into a neutron or vice versa takes placewithin a nucleus. Quantum processes are described by unitary operators,which perform the change continually via the intermediate states that aresuperpositions of proton and neutron states. This is all very natural in thenucleon description and impossible in the proton-neutron one, where a so-called super-selection rule, prohibiting the mentioned intermediate states, isvalid.In the epistemological scheme on how to improve our approximation ofreality (see the Introduction) the last passages describe the ’more’, the im-provement that the nucleon theory yields in comparison with the proton-neutron theory. Hence, we can consider that it describes reality better , i.e., that it is a better approximation to reality as far as the particles makingup the nuclei are concerned. To begin with, let us consider a short historical approach.The inventor of the exclusion principle, Pauli, is reported to have said(private communication by the late Rudolf E. Peierls) that if two electronsare apart, then they are distinct particles by this very fact. His principleapplies to those that are not in this relation.Let us make possible a concrete discussion of fermions being apart.Let D e be a spatial domain comprising the earth, and D out the com-plementary domain (in the set-theoretical sense, within all space). The twodomains are, of course, disjoint from each other. The single-fermion distin-guishing projectors are Q i ≡ Z Z Z D i | ~r ih ~r | d ~r, i = e,out . (10)They are orthogonal to each other due to the disjointness of the domains,and Q e + Q out = I , where I is the identity operator.Generalizing Pauli, Schiff (1955) stipulates that two identical particlesare distinguishable when the two-particle probability amplitude a (1 ,
2) of9ome dynamical variable is different from zero only when the two particleshave their values in disjoint ranges of the spectrum of the variable.But, as De Muynck (1975) remarks, this actually cannot ever occur whenthe wave function is anti-symmetric, for then a (2 ,
1) = − a (1 ,
2) .Taking up Schiff’s attempt to formalize a generalization of Pauli’s dis-tinguishing two identical fermions, we assume that that Schiff’s two-particleamplitude a (1 ,
2) is a two-particle wave function. Then, we know fromtextbooks that a (2 ,
1) = − a (1 ,
2) for identical fermions.Let, further, the index value in Q ei , and Q outi , i = 1 , the correct way to express Pauli’s criterion of distinguishability is to say that the two-particle system possesses the property (cf (C.1) inAppendix C) expressed by the two-identical-fermion distinguishing projector( Q e Q out + Q out Q e ) : ( Q e Q out + Q out Q e ) a (1 ,
2) = a (1 ,
2) (11)(cf (C.1)).The general theory of distinguishing identical fermions expounded in Ap-pendix B enables one to transform effectively the distinct extrinsic properties(being on earth or outside it in our concrete example) into intrinsic ones by isomorphic transition from the subspace ( Q e Q out + Q out Q e ) A ( H ⊗H ) tothe effective distinct-particle state space (cid:16) Q e H ⊗ Q out H (cid:17) . Schiff’s men-tioned criterion is actually valid in the latter, distinct-particle space.Mirman’s (1973) claim of the essential role played by experiments showsup in the fact that the mentioned transformation of extrinsic properties intoeffective intrinsic ones is restricted to experiments in which the possession ofthe distinguishing property ( Q e Q out + Q out Q e ) is preserved.Thus, a generalized Pauli criterion of distinguishing identical particlescan be expressed in the quantum-mechanical formalism quite satisfactorilyas far as two identical particles are concerned.We see that Pauli’s idea of two fermions being ”apart”, which is, nodoubt, a spatial idea, should not be understood in the sense of distance (of’far apart’), but only in the sense of disjoint domains.Since disjoint-domain distinction is completely analogous to the proton-neutron difference, there is, obviously, no problem in extending the formerdistinction to any number of fermions , in particular, to all fermions in10he universe and to any number of disjoint domains, along the lines of thegeneral ’distinct-identical-particle theory’ of Appendix B. But there is animportant difference in the two distinctions discussed (see subsection 4.1).In all experiments done in the laboratories on earth, the relevant observ-ables satisfy the required restrictions of compatibility with the correspondingdistinguishing property. But one must wonder if all important observablescan be measured within earthbound laboratories. (For a negative answer seesubsection 4.1 .)Anti-symmetrizing all fermions of a kind in the universe gives ’more’ inprincile (cf the Introduction) than Pauli’s original cautious formulation forseveral reasons with evident physical meaning:(a) One assumes as little as possible (Occham’s razor - the demand toeconomize in assumptions).(b) The formalism does not favor space over other observables. Namely,it is clear that distinction in terms of disjoint domains in the spectrum ofany other observable (or set of compatible observables) can take the place ofthe position observable. Thus, there is no need to find justification for theunique conceptual position of space in quantum mechanics.Thus, according to the epistemological scheme advocated in the Intro-duction, universal anti-symmetrization for any kind of identical fermions iscloser to reality then doing it in separate domains. In other words, we obtaina better theoretical approximation to reality in the described manner. In this article a firm attitude is taken that there exists a quantum realityindependent from the observer, and that we approach it with our theorieslike one approximates the irrationals on the real axis by rationals becausethe former can never be expressed exactly. The main point is that we canimprove the approximation, i. e., make a better theory as explained in theIntroduction. An example of such improvement is given in Appendix Bfor identical fermions. (It is a general theory how to distinguish identicalfermions or bosons, but it is only a particular example as far as improving atheory is concerned.)To make it more comprehensible, it is shown in some detail in section11 that the concept of nucleons brings us closer to reality then the idea ofprotons and neutrons does. This concrete example of the identical-fermiontheory in Appendix B has been called the nucleon paradigm because it isviewed as a basic example for the general scheme of making a better theory.Also the exclusion principle is discussed (in section 3) from the point of viewof Appendix B and the scheme of how to improve a theory.It is desirable to shed additional light on some salient features of the twocases of distinguishing identical fermions. We are interested in differencesbetween some features of nucleons and and analogous features of fermionsin disjoint spacial domains. We also want to have another look at the con-version of intrinsic into extrinsic particle properties, and vice versa , whichmakes the physical basis of the entire distinguishing theory.Concerning the effective distinct-cluster description in Appendix B (wehave two clusters both in the case of protons and neutrons and in our ex-ample of disjoint spatial domains), one should note that it is not an ap-proximation (as effective particles often are). For states that possess thedistinguishing property and for observables that are compatible with it, thedescription is exact , and for those that do not possess it (are not compatiblewith it) it does not make sense .We saw that Pauli himself mentioned ’being spatially apart’ in the formu-lation of his principle. In case of the nuclear particles, his exclusion principlewas also articulated separately for protons and separately for neutrons.All this is not wrong, but it has turned out that one can do better, andthus make a theory that approximates reality more closely (see the Introduc-tion).
The barion family discussed in section 2 consists of nuclei, and no superpo-sitions of distinct nuclei in the same family are observed in nature. Sincethe many-nucleon distinguishing property requires precisely this, all nucleipossess the many-nucleon distinguishing properties with different number ofprotons N pr .This is not so in the case of spatial disjoint-domain distinction for some12ind of fermions, e. g., nucleons. Particles can be, and often are, delocalized spatially. If described by a wave function, it is a superposition of a component(wave function) that is in the domain of earth, and one that is outside. (Likein the case of passing a double slit, when the delocalized photons or massiveparticles that pass both slits simultaneously are the object of experimenting.)Delocalized particles do not possess the many-particle distinguishing prop-erty, and hence they cannot be treated separately on earth, and separatelyoutside earth. They must be omitted from the effective distinct-particle de-scription. In this sense, the latter theory approximates the wider identical-fermion one even where the many-particle distinguishing property is ob-served. The fewer fermions are left out, the better the description.One wonders if there is anything wrong with applying quantum mechan-ics to a restricted domain, re. g., earth or a laboratory on earth. The answeris ”yes”. We give an argument against the exactness of local quantum me-chanics of this kind.When the orbital (or spatial) tensor-factor space of a single particle isdetermined by the basic set of observables, which are the position, the linearmomentum, and their functions, spin etc., one obtains an irreducible space,i. e., a space that has no non-trivial subspace invariant simultaneously for allthe basic observables (for position and linear momentum; cf sections 5 and6 in chapter VIII of Messiah’s (1961) book. Hence, the above used subspace Q e H (for the local, earth quantum-mechanical description) is not invarianteither. It is, of course, invariant for position, but linear momentum has tobe replaced by another Hermitian operator approximating it. vice versa As it was stated, the notion of identical particles rests on the idea of equal in-trinsic properties of the particles. One can view the general theory expoundedin Appendix B as the general framework how to convert some extrinsic prop-erties , represented by nontrivial projectors in the single-particle state space, into intrinsic ones . The extrinsic properties are converted into intrinsic onesin terms of single-particle distinguishing projectors { Q j : j = 1 , , . . . , J } generating the distinct clusters (cf (B.1) and (B.2)). In the effective distinct-cluster space H D ...N (cf (B.3a-c)) these properties become actually intrinsic.13t is important to notice that the effective distinct-cluster space H D ...N is still expressed by projectors. But since the description is restricted totheir ranges (one is within the space), they amount to the same a intrinsicproperties.One should also pay attention to the difference in our two two-cluster the-ories. In the nucleon paradigm one could even get rid of the single-nucleondistinguishing projectors Q + and Q − by eliminating the isospin tensorfactor space in the single-nucleon space (1a) and going over to the protonspace (1b) and the neutron space (1c). In the disjoint-spatial-domain dis-tinction there is no suitable way to do something analogous. But when onerestricts the description to the (invariant) range of the many-distinct fermionspace H D ...N , this simulates the convertion of the extrinsic property into anintrinsic one in a satisfactory manner.Sometimes the reverse conversion of intrinsic properties into extrinsicones takes place. For this algorithm the same conceptual framework fromAppendix B can be used. In other words, the theory presented in this articlecovers also this case.The best example is that of protons and neutrons, where historically(and in many textbooks) the proton-neutron description is given priority ifnot presented exclusively.The reverse process at issue consists in transferring the quantum-mechanicaldescription from H D ...N to the subspace H id ...N of the first-principle space A ...N Q ⊗ ,N nu n =1 H n . Inclusion of β -radioactivity requires the use of the latterspace because that of the former does not suffice.Perhaps additional light is shed on the reverse application if the ex-pounded theory by discussing a fictitious case . Suppose we want to treatthe proton (pr) and the electron (el) as two states of a single particle (likethe proton and the neutron). Can we do this? The answer is affirmative, andthe way to do it is to use the theory of this article in the, above explained,reverse direction.The new first-particle space would be H ≡ Q pr H ⊕ Q el H , where Q pr and Q el project H onto the proton and the electron subspace re-spectively. The rest is analogous to the case of the nucleons in section 2with the important difference that there is no counterpart of the effect of theweak interaction. This means that every realistic N -particle state ρ id ...N H D ...N will always do for description, andthe simplicity requirement (the razor of Occham) brings us back to perma-nently distinct particles.Actually, one speaks of identical particles if the particles have identical complete sets of intrinsic properties.This condition has the prerequisite that long experience suggests that oneis unable to convert any of the intrinsic properties by dynamical means intoextrinsic ones, and that one is unable to extend the set of such properties.These are impotency stipulations analogous to those of thermodynamics onwhich the thermodynamical principles are based.Let a good illustration be given for this. Some time ago the electron neu-trino and the muon neutrino were believed to be identical particles becausethey had their, up-to-then known, intrinsic properties in common. Laterit was discovered that they differ; the former has the electronic leptonicquantum number, and the latter the muonic one. Thus, their other com-mon properties were incomplete; after completion it turned out that they nolonger have all intrinsic properties equal.An illustration for converting an intrinsic property into an extrinsic oneis the case of parity and weak interaction. Until the advent of the famousparity-non-conserving weak interaction experiments, parity could be consid-ered an intrinsic property of the elementary particles. These experimentsconverted it into an extrinsic one, and nowadays we must work with the par-ity observable with its parity-plus and parity-minus eigen-projectors. Appendix A. The necessary textbook formalism and the anti-symmetrizer
This Appendix has the purpose to remind the reader of thebasic first-quantization (as distinct from second-quantization) textbook no-tions for the treatment of identical particles.One has N single-particle state spaces {H n : n = 1 , . . . , N } . The identicalness of the particles is expressed (i) in terms of isomorphisms { I m → n : m, n = 1 , . . . , N ; m = n } mapping the single-particle space H m onto H n , m, n = 1 , , . . . , N m = n . Naturally, I m → n I n → m = I n , I n being the identity operator in H n , n = 1 , , . . . , N .Any two equally-dimensional Hilbert spaces are isomorphic, and there are15ery many different isomorphisms connecting them. For the identicalness themore important requirement is the following requirement on the isomorphismin (i): (ii) the physically meaningful operators, observables in particular, ineach single-particle space are equivalent with respect to the isomorphismsgiven in (i).As an illustration, we mention that the second-particle radius-vector op-erator, e. g., is: ~r = I → ~r I → . The
N-distinct-particle space , on which the description of identicalparticles in first-quantization quantum mechanics is based, is H ...N ≡ ⊗ ,N Y n =1 H n , ( A. ⊗ denotes the tensor (or direct) product of Hilbert spaces. (Weshall use this symbol also for the tensor product of vectors and of operators.)The anti-symmetrizer (for identical fermions), written as a projector, is A ...N ≡ ( N !) − X p ∈S N ( − p P ...N . ( A. S N is the so-called symmetric group, i. e., the group of all N !permutations of N objects, in this case of N identical particles, by p are denoted the elements of the group, ( − p is the parity of the permuta-tion. It is +1 if the permutation can be factorized into an even number oftranspositions, and it is − Q ⊗ ,Nn =1 H n are deter-mined with the help of the isomorphisms { I m → n : m, n = 1 , . . . , N ; m = n } defined above.Finally, P ...N are the unitary operators that represent the permuta-tions p in the N-distinct-particle space given by (A.1). When acting on anuncorrelated vector, P ...N permutes the tensor factor single-particle statevectors according to the prescription contained in p .It should be noted that the state space of N identical fermions is A ...N H ...N ( A. ppendix B. How to obtain distinct identical fermions We utilize the powerful tool of projectors and elementary group theory.
In the general case , which we are now going to elaborate, let the distin-guishing events (or properties), which are going to generate the distinctnessof the identical particles, be given by J orthogonal single-particle pro-jectors : {{ Q jn : j = 1 , . . . , J } : n = 1 , . . . , N } , ∀ j, ∀ n : ( Q jn ) † = Q jn (Hermitian operators), ∀ n : Q jn Q j ′ n = δ j,j ′ Q jn (orthogonal projectors), andfinally, ∀ j : Q jn = I → n Q j I n → , n = 2 , . . . , N (mathematically, equivalentprojectors; physically, same events or properties).We have in mind J clusters of effectively-distinct particles , 2 ≤ J ≤ N. We enumerate them by j in an ordered way according to the(arbitrarily fixed) values of j : j = 1 , . . . , J. The j -th cluster containsa certain number of particles, which we denote by N j , P Jj =1 N j = N. Itwill prove useful to introduce also the sum of particles up to the beginningof the j -th cluster: M j ≡ P ( j − j ′ =1 N j ′ for j ≥ , and M ≡ . The single-particle distinguishing projectors appear in the distinct-particlespace H ...N (cf (A.1)) through a tensor product determining the N -particle distinguishing projector Q ...N in H ...N : Q ...N ≡ ⊗ ,J Y j =1 (cid:16) ⊗ , ( M j + N j ) Y n =( M j +1) Q jn (cid:17) . ( B. j -th cluster ,and it consists of the tensor product of physically equal (mathematicallyequivalent via transpositions) single-particle projectors (and there are J clusters).We want to introduce the corresponding distinct-cluster space H D ...N , which is the state space consisting of the tensor product of J distinct-particle clusters (see (B.3a) and (B.3b) below) , each consisting of identicalparticles, and hence anti-symmetrized.Let us call the ’cluster subgroup’ , and denote by S clN , the subgroupof permutations of N objects that act possibly nontrivially only within thegiven clusters. (It is a subgroup of S N , the group of all (N!) permutations of N ’cluster anti-symmetrizer’ , we denote itby A cl ...N , is of the form A cl ...N ≡ ⊗ ,J Y j =1 A ( M j +1) ... ( M j + N j ) = X p ∈ S clN ( − ) p P ...N . J Y j =1 ( N j !) , ( B. H D ...N is defined as follows H D ...N ≡ ⊗ ,J Y j =1 h A ( M j +1) ... ( M j + N j ) (cid:16) ⊗ , ( M j + N j ) Y n =( M j +1) ( Q jn H n ) (cid:17)i = n ⊗ ,J Y j =1 h A ( M j +1) ... ( M j + N j ) (cid:16) ⊗ , ( M j + N j ) Y n =( M j +1) Q jn (cid:17)io H ...N = Q ...N A cl ...N H ...N , ( B. a, b, c )where a, b, c refer to the three obviously equivalent expressions determining H D ...N . (and the two operator factors in (B.3c) commute). Here by A withindices running within one cluster are denoted the cluster anti-symmetrizers(cf (B.2)).Note that the individual distinct cluster spaces (factors in the tensorproduct Q ⊗ ,Jj =1 in (B.3a)) are decoupled from each other (in the sense ofidentical-fermion symmetry correlations), i. e., one has the tensor product Q ⊗ ,Jj =1 , but the factor spaces within each cluster are coupled by the corre-sponding anti-symmetrizers.The order of the distinguishing projectors Q jn in (B.1), within the clus-ters and of the clusters, is mathematically arbitrary and physically irrele-vant. Hence, in view of the fact that we are dealing with identical fermionsin A ...N H ...N , the suitable entity is not Q ...N given by (B.1). It is the symmetrized distinguishing projector Q sym ...N obtained from (B.1) bythe permutation operators: Q sym ...N ≡ (cid:18) X p ∈S N (cid:16) P ...N Q ...N P − ...N (cid:17)(cid:19). J Y j =1 ( N j !) . ( B. distinguishing property , expressed as the symmetric pro-jector (B.4) in H ...N , which commutes with every permutation operator P . Hence, it commutes with the anti-symmetrizer A ...N (which is a lin-ear combination of permutation operators cf (A.2)) and it reduces in theidentical-fermion space A ...N H ...N .Of central importance is the following N -identical-fermion subspace H id ...N of A ...N H ...N . It is defined as the range of Q sym ...N in the identical-fermion state space A ...N H ...N : H id ...N ≡ Q sym ...N A ...N H ...N = A ...N Q sym ...N H ...N . ( B. heorem 1. The isomorphisms.
The subspaces H id ...N and H D ...N of H ...N are isomorphic , and the operators in H ...N I id → D ...N ≡ (cid:16) ( N !) . J Y j =1 ( N j !) (cid:17) / Q ...N , ( B. I D → id ...N ≡ (cid:16) ( N !) . J Y j =1 ( N j !) (cid:17) / A ...N ( B. mutually inverse unitary isomorphisms mapping H id ...N onto H D ...N and vice versa: H D ...N = I id → D ...N H id ...N and H id ...N = I D → id ...N H D ...N . Proof.
First , we show that Q ...N maps H id ...N into H D ...N .One should note that Q sym ...N is an orthogonal sum of N ! . Q Jj =1 N j !(the number of cosets of the cluster subgroup S clN in S N ) projectors in H ...N (cf (B.1)). As a consequence, the subprojector relation Q ...N ≤ Q sym ...N is valid: Q ...N Q sym ...N = Q ...N . ( B. P ...N is a permutation the action of which is not restricted to withinthe given clusters, then Q ...N P ...N H ...N = Q ...N ( Q ...N P ...N H ...N ) = 0because in the subspace in the brackets at least one single-particle distin-guishing projector appears outside its cluster, and Q ...N (cf (B.1)) actingon it gives zero. Hence, in view of the definition of A ...N (cf (A.2)), anddue to the fact that the ’cluster anti-symmetrizer’ is of the form (B.2), onehas Q ...N A ...N = ( N !) − J Y j =1 N j ! Q ...N A cl ...N . ( B. Q ...N with A cl ...N (cf (B.1) and(B.2)) as well as (B.8), definition (B.3c) finally gives for every element in | Ψ i ...N ∈ H ...N : Q ...N A ...N Q sym ...N | Ψ i ...N = ( N !) − J Y j =1 N j ! Q ...N A cl ...N | Ψ i ...N ∈ H D ...N (cf (B.3c)). Thus, I id → D ...N (cf (B.6)) maps H id ...N into H D ...N as claimed.19or the proof in the opposite direction , one should note that one also hasthe evident subprojector-relation A ...N ≤ A cl ...N : A ...N A cl ...N = A ...N . ( B. | Φ i ...N ∈H ...N (cf (B.3c)): A ...N Q ...N A cl ...N | Φ i ...N =( N !) (cid:16) J Y j =1 N j ! (cid:17) − A ...N (cid:16) A ...N Q ...N A ...N ) A cl ...N | Φ i ...N . On account of the relations ∀ p ∈ S N : A ...N P ...N = P − ...N A ...N = ( − ) p A ...N , ( B. A ...N Q sym ...N A ...N = (cid:16) N ! . J Y j =1 N j ! (cid:17) A ...N Q ...N A ...N . ( B. A ...N Q ...N A cl ...N | Φ i ...N = (cid:16) N ! . J Y j =1 N j ! (cid:17) − A ...N Q ...N A ...N | Φ i ...N = Q sym ...N A ...N | Φ i ...N ∈ H id ...N as claimed (cf (B.3c) and (B.5)). The last step has take into account thecommutation of A ...N with Q sym ...N and the idempotency of the former. Next , we show that the maps I id → D ...N and I D → id ...N in application to thesubspaces H id ...N and to H D ...N respectively are each other’s inverse.Owing to the definitions (B.6) and (B.7), and to the definition (A.2) of A ...N one has the following equality of maps: I id → D ...N I D → id ...N = n(cid:16) ( N !) . J Y j =1 ( N j !) (cid:17) / Q ...N (cid:16) ( N !) . J Y j =1 ( N j !) (cid:17) / A ...N =20 (cid:16) J Y j =1 ( N j !) (cid:17) − X p ∈S N ( − ) p Q ...N P ...N o . For any | Φ i ...N = Q ...N A cl ...N | Φ i ...N ∈ H D ...N (cf (B.3c)), the definition(A.2) of the anti-symmetrizer implies I id → D ...N I D → id ...N | Φ i ...N = n(cid:16) J Y j =1 ( N j !) (cid:17) − X p ∈S N ( − ) p Q ...N P ...N Q ...N o A cl ...N | Φ i ...N = (cid:16) J Y j =1 ( N j !) (cid:17) − X p ∈S N Q ...N (cid:16) P ...N Q ...N P − ...N (cid:17)(cid:16) ( − ) p P ...N A cl ...N (cid:17) | Φ i ...N . All (cid:16) P ...N Q ...N P − ...N (cid:17) multiply with Q ...N into zero except when p ∈ S clN , and then P ...N Q ...N P − ...N = Q ...N . Hence, the sum gives thisprojector (cid:16) Q Jj =1 ( N j !) (cid:17) times. Besides, for p ∈ S clN , P ...N commuteswith A cl ...N (cf (B.2)) and (cid:16) ( − ) p P ...N A cl ...N (cid:17) = A cl ...N by an elementaryargument analogous to that giving the adjoint of (B.11). Thus, finally, I id → D ...N I D → id ...N | Φ i ...N = Q ...N A cl ...N | Φ i ...N = | Φ i ...N . This establishes the claim that I id → D ...N is the inverse of I D → id ...N . Analogously , in view of (B.6) and (B,7), we have the following equality ofoperators due to (A.2): I D → id ...N I id → D ...N = (cid:16) ( N !) . J Y j =1 ( N j !) (cid:17) / A ...N (cid:16) ( N !) . J Y j =1 ( N j !) (cid:17) / Q ...N = (cid:16) J Y j =1 ( N j !) (cid:17) − X p ∈S N (cid:16) P ...N Q ...N P − ...N (cid:17) ( − ) p P ...N . for any | Ψ i ...N = A ...N Q sym ...N | Ψ i ...N ∈ H id ...N , one can write I D → id ...N I id → D ...N | Ψ i ...N = (cid:16) J Y j =1 ( N j !) (cid:17) − X p ∈S N (cid:16) P ...N Q ...N P − ...N (cid:17) ( − ) p P ...N A ...N | Ψ i ...N =21 sym ...N A ...N | Ψ i ...N . In the last step we have utilized the adjoint of (B.11) and the definition (B.4).Thus, the claim that the two maps are the inverse of each other is proved.Since the maps are the inverse of each other, it is easily seen hat they arenecessarily surjections and injections, i. e., bijections as claimed.
Next , we prove that I D → id ...N preserves the scalar product, which wewrite as (cid:16) . . . , . . . (cid:17) . Let Ψ ...N and Φ ...N be two arbitrary elementsof H D ...N . On account of (B.7) one has (cid:16) I D → Id ...N Ψ ...N , I D → Id ...N Φ ...N (cid:17) = h ( N !) . J Y j =1 ( N j !) i(cid:16) Ψ ...N , A ...N Φ ...N (cid:17) . Further, on account of the fact that both Q ...N and A cl ...N acts as theidentity operator on Ψ ...N and Φ ...N (cf (B.3c)), one can write lhs = (cid:16) J Y j =1 ( N j !) (cid:17) − X p ∈S N (cid:16) Ψ ...N , Q ...N ( P ...N Q ...N P − ...N ) (cid:16) ( − ) p P ...N A cl ...N Φ ...N (cid:17) . Again one has Q ...N ( P ...N Q ...N P − ...N ) = 0 , except if p ∈ S clN , when it isequal to Q ...N . Therefore, lhs = (cid:16) Ψ ...N , Q ...N A cl ...N Φ ...N (cid:17) = (cid:16) Ψ ...N , Φ ...N (cid:17) as claimed.It is easy to see that also the inverse of a scalar-product preserving bijec-tion must be scalar-product preserving. This completes the proof of Theorem1.
The physical meaning of the identical-fermion symmetry decoupling andthe coupling isomorphisms I id → D ...N and I D → id ...N respectively given in thetheorem shows up primarily, of course, in the observables that are defined in H id ...N and H D ...N . The corresponding or equivalent operators (obtainedby the similarity transformation) are of the same kind: Hermitian, unitary,22rojectors etc. because all these notions are defined in terms of the Hilbert-space structure, which is preserved by the (unitary) isomorphisms.It is seen that a prerequisite for describing an evolution or a measurementin the subspaces H id ...N and H D ...N is the possession of the distingwishingproperties (occurrence of the events) Q sym ...N and Q ...N respectively, andtheir preservation.A relevant observable for the decoupling, i. e., a Hermitian operator thatreduces in H id ...N , is one that commutes with the distinguishing projector Q sym ...N , and one confines oneself to its reducee in H id ...N . In physical terms,the observable must be compatible with the distinguishing property Q sym ...N and one must assume that the property is possessed (cf (C.1) and (C.2) inAppendix C), and that this is preserved if some process is at issue. Theorem 2. A)
Let O D ...N be an operator in H ...N (with a physi-cal meaning) that commutes (is compatible) both with every permutationoperator permuting possibly non-trivially only within each cluster (’clusterpermutations’) and with Q ...N (cf (B.1)). (Hence, O D ...N reduces in H D ...N , cf (B.3c).) Let, further, O id, ( D )1 ...N ≡ (cid:16) J Y j =1 ( N j !) (cid:17) − X p ∈S N P ...N O D ...N Q ...N P − ...N ( B. O D ...N and Q ...N . Then O id, ( D )1 ...N com-mutes with every permutation operator (it is a ’symmetric’ operator), hencewith A ...N (cf (A.2)), and with Q sym ...N (cf (B.4)), and the reducee of O D ...N in H D ...N (cf (B.3c)) and that of O id, ( D )1 ...N in H id ...N (cf (B.5))respectively are equivalent (physically the same observables) with respectto the isomorphisms in Theorem 1. One can express this in H ...N by theoperator equality: O id, ( D )1 ...N A ...N Q sym ...N = (cid:16) I D → id ...N (cid:17) O D ...N (cid:16) I id → D ...N (cid:17) A ...N Q sym ...N ( B. a )(cf (B.5)). B) Conversely, let B id ...N be a symmetric operator in H ...N (with aphysical meaning) that commutes (is compatible) with Q sym ...N . Then theoperator B D, ( id )1 ...N ≡ Q ...N B id ...N Q ...N ( B. b )23in H ...N ) commutes with every cluster permutation, hence with A cl ...N (cf (B.2)), and with Q ...N (cf (B.1)). The reducee of B D, ( id )1 ...N in H D ...N is equivalent with (physically the same observable as) the reducee of B id ...N in H id ...N . The equivalence is given by the operator relation in H ...N : B D, ( id )1 ...N A cl ...N Q ...N = (cid:16) I id → D ...N (cid:17) B id ...N (cid:16) I D → id ...N (cid:17) A cl ...N Q ...N ( B. c )(cf (B.3c)). Proof. A)
Since ∀ p ′ ∈ S N , also { P ′ ...N P ...N : ∀ p ∈ S N } is the sym-metric group (so-called translational invariance of groups), one has ∀ p ′ ∈ S N : P ′ ...N O id, ( D )1 ...N ( P ′ ...N ) − = (cid:16) J Y j =1 ( N j !) (cid:17) − X p ∈S N ( P ′ ...N P ...N ) O D ...N ( P ′ ...N P ...N ) − = O id, ( D )1 ...N , which is obviously equivalent to commutation of the operator with everypermutation operator. Hence it commutes also with A cl ...N (cf (B.2)).On account of the facts that both Q ...N and O D ...N commute withevery cluster permutation, we choose arbitrarily one permutation p k fromeach coset of S clN in S N , i. e., we view S N as the set-theoretical sum ofcosets S N = P Kk =1 p k S clN , where K = N ! . Q Jj =1 N j ! . Then we can write Q sym ...N = K X k =1 P k ...N Q ...N ( P k ...N ) − (15)(cf (B.4)), and O id, ( D )1 ...N = K X k =1 P k ...N Q ...N O D ...N Q ...N ( P k ...N ) − (16)(cf (B.13)). Further,( P k ...N Q ...N ( P k ...N ) − )( P k ′ ...N Q ...N ( P k ′ ...N ) − =( P k ′ ...N Q ...N ( P k ′ ...N ) − )( P k ...N Q ...N ( P k ...N ) − = 0 if k = k ′ (cf (B.1)). Therefore, Q sym ...N O id, ( D )1 ...N = O id, ( D )1 ...N Q sym ...N = O id, ( D )1 ...N .
24e thus have commutation of O id, ( D )1 ...N with Q sym ...N , and the former oper-atorn reduces in H id ...N (cf (B.5)).The claimed relation (14a) is explicitly: O id, ( D )1 ...N A ...N Q sym ...N = h ( N !) .(cid:16) J Y j =1 ( N j !) (cid:17)i A ...N O D ...N Q ...N A ...N Q sym ...N . ( B. B.
12) , one has A ...N O D ...N Q ...N A ...N = h ( N !) .(cid:16) J Y j =1 ( N j !) (cid:17)i − A ...N O id, ( D )1 ...N A ...N . Replacing this in rhs(B.17), one obtains rhs ( B.
17) = A ...N O id, ( D )1 ...N A ...N Q sym ...N . Finally, on account of commutation of the symmetric operator O id, ( D )1 ...N withthe (idempotent) projector A ...N (cf (A.2)), rhs ( B.
17) = lhs ( B.
17) . B) Since B id ...N is by assumption a symmetric operator and Q ...N commutes with each cluster permutation operator (cf (B.1)), so does B D, ( id )1 ...N .Hence, B D, ( id )1 ...N commutes also with A cl ...N (cf B.2)). The former operatorobviously commutes with the (idempotent) projector Q ...N . Therefore itreduces in H D ...N (cf (B.3c)).In view of (B.14b), the claimed relation (B.14c) has the following explicitform Q ...N B id ...N Q ...N A cl ...N Q ...N = h ( N !) .(cid:16) J Y j =1 ( N j !) (cid:17)i Q ...N B id ...N A ...N A cl ...N Q ...N . ( B. Q ...N and A cl ...N and on account of the adjoint of (B.9). This ends the proof.
In case the state (density operator) ρ id ...N of an N -identical-fermionsystem satisfies the relation Q sym ...N ρ id ...N = ρ id ...N , ( B. possesses the distinguishing property Q sym ...N in thestate in question (cf (C.2) in Appendix C). In this case, and only in this case,it is amenable to Theorem 2.It is important to notice that the theory presented in this Appendix isapplicable to identical bosons equally as to identical fermions, one muist onlyreplace the anti-symmetrizer projector A ...N (cf (A.2)) by the symmetrizerprojector S ...N ≡ P p ∈ S N P ...N . N ! .This theory was presented in more detail and in a form valid simultane-ously for fermions and bosons in (Herbut 2006). Also much relevant mathe-matical help was included, especially about the symmetric group. But read-ing it, there is a price to pay: the exposition is less readable. Appendix C. Possessed property or event that has occurred
Let E be a projector (physical meaning: property or event). A) Pure-state case.
Let | ψ i be a state vector (pure state). Then h ψ | E | ψ i = 1 ⇔ E | ψ i = | ψ i , ( C. ⇔ ” denotes logical implication in both directions. Proof. h ψ | E | ψ i = 1 ⇔ h ψ | E ⊥ | ψ i ) = 0 , where E ⊥ ≡ I − E , I being the identity operator, and E ⊥ is theortho-complementary projector. Further, h ψ | E | ψ i = 1 ⇔ || E ⊥ | ψ i|| = 0 ⇔ E ⊥ | ψ i = 0 ⇔ E | ψ i = | ψ i . ✷ B) General-state case.
Let ρ be a density operator (general state).Then tr( Eρ ) = 1 ⇔ Eρ = ρ. ( C. Proof.
Let ρ = P i r i | i ih i | be a spectral form of ρ in terms of itseigen-vectors {| i i : ∀ i } corresponding to its positive eigenvalues { r i : ∀ i } .(It always exists.) Thentr( Eρ ) = 1 ⇔ X i r i tr( E | i ih i | ) = 1 ⇔ X i r i h i | E | i i = 1 ⇔ i r i (1 − h i | E | i i ) = 0 ⇔ ∀ i : h i | E | i i = 1 ⇔ ∀ i : E | i i = | i i ⇔ ∀ i : Eρ = ρ. In the last but one step (C.1) has been made use of. ✷ References
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