arXiv:2105.04288v1 [quant-ph] 10 May 2021 eeaiaino h n-iesoa oo-emo Dua - One-Dimensional the of Generalization A hrceie by t characterized and of model construction explicit Lieb-Liniger an the present parti pat between indistinguishable the duality to using boson-fermion the approach By configuration-space contacts. the two-body as through only interacting cles cl-nain eeaiaino h oo-emo duality. boson-fermion the strong of the generalization and scale-invariant mapping, boson-fermion the equivalence, spectral esuybsnfrindaiisi n-iesoa aybd p many-body one-dimensional in dualities boson-fermion study We 1 − n ad-uuaa --4 hyd,Tko1180,Japan 101-8308, Tokyo Chiyoda, 1-8-14, Kanda-Surugadai hog h ahItga Formalism Path-Integral the Through itnt(oriaedpnet opigcntns hs model These constants. coupling (coordinate-dependent) distinct nttt fQatmSine io University, Nihon Science, Quantum of Institute [email protected] n bsnand -boson Dtd a 1 2021) 11, May (Dated: aoh Ohya Satoshi Abstract n 1 frinmdl hc r ult ahohrand other each to dual are which models -fermion ls efidagnrlzto of generalization a find we cles, wa ult.W lodsusa discuss also We duality. -weak eCenSieaamdl We model. Cheon-Shigehara he -nerlfraima well as formalism h-integral olm fietclparti- identical of roblems no the enjoy s lity 1 Introduction

Inhisseminalpaper[1] in 1960, Girardeau proved the one-to-one correspondence—the duality—between one-dimensional spinless and with hard-core interparticle . By using this duality, he presented a celebrated example of the spectral equivalence between impenetrable bosons and free fermions. Since then, the one-dimensional boson-fermion duality has been a testing ground for studying strongly-interacting many-body problems, especially in the field of integrable models. So far there have been proposed several generalizations of the Girardeau’s finding, the most promi- nent of which was given by Cheon and Shigehara in 1998 [2]: they discovered the fermionic dual of the Lieb-Liniger model [3] by using the generalized pointlike interactions. The duality between the Lieb-Liniger model and the Cheon-Shigehara model is a natural generalization of [1] and consists of (i) the spectral equivalence between the bosonic and fermionic systems, (ii) the one-to-one mapping between bosonic and fermionic wavefunctions, and (iii) the one-to-one correspondence between a strong-coupling regime in one system and a weak-coupling regime in the other (i.e., the strong-weak duality). The purpose of the present paper is to derive and further generalize this duality by using the path-integral formalism. Before going into details, however, let us first briefly recall the basics of the boson-fermion duality by simplifying the argument of [2]. Let us consider n identical which have no internal structures (i.e., spinless), move on the whole line R, and interact through two-body contact interactionsbut otherwise freely propagate in the bulk. The Lieb-Liniger model and the Cheon-Shigehara model are particular examples of such n-body systems and described by the following Hamiltonians, respectively:

2 n 2 2 ℏ ) ℏ 1 (1a) HB = − É 2 + É (xj − xk; a ) 2m j=1 )xj m 1≤j

¨ 2  Notice that B and its derivatives become continuous in the limit 1/ 0, which indicates that 1/ describes the deviation from the smooth continuous theory (i.e., the free theory)and plays the role of a coupling constant in the Lieb-Liniger model (1a). In contrast, F and itsa derivatives → become continuousa in the limit 0, which indicates that describes the deviation from the free theory and plays the role of a coupling constant in the Cheon-Shigehara model (1b ). This inverse relation of the coupling constants is already incorporated into the notations 1 and and the heart of the strong-weak a → a ( ; a ) ( ; ) duality. Now, all the above connection conditions are valid x for generic" x waavefunctions without any symme- try. However, further simplifications occur if the wavefunctions are totally symmetric (antisymmetric) under the exchange of coordinates, which must hold for identical spinless bosons (fermions) due to the indistinguishability of in . For example, if B is the totally symmetric function and satisfies the identity B( j k ) = B( k j ), there automat- ically hold the additional conditions and ) B ) B , which reduce (2a) B x =0 = B x =0 )x x =0 = − )x x =0 to ) B 1 . Similarly, if is⋯ the , x totally,⋯,x ,⋯ antisymmetric ⋯ , x ,⋯,x function,⋯ and satisfies the )x x =0 − a B x =0 = 0 jk + F jk − jk jk + jk jk − | | ó ó identity F( j ) = − F( j ), there automaticallyó hold theó additional conditions jk jk + jk k + k ó | and ) F ) F , which reduce (3a) to ) F . Putting F x =0ó = − F x =0 )x x =0 = )x x =0 F x =0 − )x x =0 = 0 all these things⋯ , x together,,⋯,x ,⋯ one immediately ⋯ , x ,⋯,x sees,⋯ that the systems of identical bosons and fermions de- jk + jk − jk + jk ó jk + jk ó jk − jk ó jk + scribed | by (1a )| and (1b) are bothó describedó by the bulk free Hamiltonian | 0 togethera ó with the following Robin boundary conditions at the coincidence points: n H B/F 1 (4) − B/F x =0 = 0 jk x =0 ) ó jk + Since both systems are described by the sameó jk + bulk Hamiltonianó and the same boundary conditions, ó ó . they automatically become isospectral.)x Thisó is thea boson-fermion duality in one dimension, which consists of the spectral equivalence between B and F, the equivalence between the strong coupling regime of B/F and the weak coupling regime of F/B, and, as we will see in section 2.3, the one-to-one mapping between B and F. Note that the extremeH H case 0 corresponds to the simplest duality between theH impenetrable bosons and the free fermions.H Now it is obvious from the above discussion that the one-dimensa → ional boson-fermion duality just follows from simple connection condition arguments. However, it took more than thirty years to arrive at the above findings since the discovery of the simplest duality by Girardeau. One reason for this would be the lack of a systematic derivation of the dual contact interactions for identical bosons and fermions. The purpose of the present paper is to fill this gap and to present a systematic machinery for deriving (and generalizing) the boson-fermion duality. As we will see in the rest of the paper, this purpose is achieved by using the path-integral formalism as well as the configuration-space approach to identical particles [7–10]. The paper is organizedas follows. In section 2 we first present the basics of the configuration-space approach to identical particles following the argument of Leinaas and Myrheim [10]. In this approach, the indistinguishability of identical particles is incorporated into the configuration space rather than the permutation of multiparticle wavefunctions. We first see that the configuration space R  Rn of identical particles on is given by the orbit space n = ( − Δn)/ n, where Δn is the set of coincidence points of two or more particles and n the symmetric group of order . We see that thisn space has a number of nontrivial boundaries, where -body contact interactionsS take place at codimension-( − 1) boundaries. We then see that,S irrespective of the ,n two-body contact interactions are generally described by the Robin bounk dary conditions at the codimension-1  boundaries of k n. We also discuss that the boson-fermion mapping holds irrespective of the boundary conditions. In section 3 we study the Feynman kernel for identical particles from the viewpoint of path  integral. We will show that, by using the Feynman kernel on n, the boson-fermion duality between (1a) and (1b) is generalized to the duality between the models described by the Hamiltonians (49a) and (49b). In section 4 we summarize our results and discuss a scale-invariant generalization of the boson-fermion duality. Appendices A and B present proofs of some mathematical formulae.

3 2 Identical particles in one dimension

One of the principles of quantum mechanics is the indistinguishability of identical particles. There are two main approaches to implement this principle into a theory. The first is to consider the permutation symmetry of multiparticle wavefunctions. The second is to restrict the multiparticle configuration space by identifying all the permutedpoints. As we will see below, the one-dimensional boson-fermion duality is best described by the second approach. In order to fix the notations, however, let us first start with the first approach. Let us consider identical particles moving on the whole line R. Let us label each particle with R a number ∈ {1 and let j be a spatial coordinate of the th particle. Let n be a permutation of indicesn that acts on a multiparticle wavefunction 1 n as follows: j ,⋯,n} x ∈ j  ∈ S (5) n (x) = (x ,⋯,x ) ¨ where is defined as (1) (n) . Note that this definition satisfies the identity ¨  ∶¨ (x) ↦ (x), ( (1)) ( (n)) for any n. The indistinguishability of identical particles then implies thatx the¨ multiparticlex¨ ∶=( wavefunctionsx ,⋯,x ) and are physically equivalent; that( is,x) the = probability( )x (= (x densities,⋯,x for these)) configurations,  ∈ mustS be the same. Thus we have (x) (x) 2 2 (6)

In other words, and must be identical up to a phase factor. Hence, | (x)| = | (x)| . (7) (x) (x) where is a phase that may depend on .2 In addition, the identity ¨ ¨ (x) = () (x), ¨ ¨ ¨ implies that must preserve the group multiplication law for any n; that is, the() map ∈ U (1) n must be a one-dimensional unitary representation (( ofx))n. = As (( is well-)x) known, there are just two such representations of n, one is the ( totally) ( ) symmetric = ( ) representation,  ∈ (i.e.,S [B] the trivial representation) ∶ S ↦ U (1)and the other the totally antisymmetric representationS (i.e., the sign [F] representation) , both of which are simply givenS by [B] (8a) [F] (8b) () = 1, Here stands for the signature of and is given by for even permutations and () = sgn(). for odd permutations. Of course, the totally symmetric representation [B] corresponds to [F] the Bose-Einsteinsgn() statistics and the totally antisymmetric representationsgn() = 1 the Fermi-Dirac statistics. sgn(In this) way, = −1 the particle statistics is determined by the one-dimensional unitary representation of the symmetric group n. There is another, more geometrical approach to identical particles, which was introduced indepen- dently by SouriauS [7, 8] and by Laidlaw and DeWitt [9] and then thoroughly investigated by Leinaas and Myrheim [10] (see also the introduction of [14] for a nice pedagogical review). In this approach, the indistinguishability of identical particles is built into the configuration space by identifying all the permuted configurations of identical particles. More precisely, given a one-particle configuration space , the configuration space of identical particles is generally given by first taking the Cartesian product of copies of , and then subtracting the coincidence points of two or more particles3, and then identifyingX points under the actionn of every permutation. The resulting space is the orbit space 2We here assumen that X is independent of the coordinates. This assumption excludes, for example, the exchange relations in [11, 12]. For simplicity, we will not touch upon coordinate-dependent particle-exchange phases in the present paper. Note that a path-integral () approach to one-dimensional was discussed in [13]. 3This subtraction procedure is the heart of the configuration-space approach and is based on the assumption that two or more particles cannot occupy the same point simultaneously [9]. This assumption may be arguable, but it successfully leads to the braid-group statistics in two dimensions and the Bose-Fermi alternative in three and higher dimensions.

4  n n n n n, where is the Cartesian product of and n the set of coincidence points. In this setting, a particle exchange is no longer described by the permutation symmetry of multiparticle  wavefunctions= (X −Δ because)/S all theX permuted configurations areX identifiedΔ in n. Instead, it is described by  dynamics: identical particlesare said to be exchanged if they start from an initial point in n and then  return to the same point in n in the course of the time-evolution. Clearly, such an exchange process  corresponds to a closed loop in the configuration space. In addition, if n is a multiply-connected space, multiparticle wavefunctions may not return to itself but rather acquire a phase after complet- ing the loop. It is such a phase that determines the particle statistics. Moreover, it can be shown that such a phase must be a member of a one-dimensional unitary representation of the fundamental group  Rd 1 n . A typical example is the case with ≥ , in which the fundamental group is n. Therefore, in three and higher dimensions, particle-exchange phases must be or , thus reproduc- R2 ing ( the) Bose-Fermi alternative. AnotherX typical= exampled is th3e case , in which the fundamentalS group becomes the braid group n. Since there is a one-parameter family of one-dimensional+1 −1 unitary i representations of n, the particle-exchange phase must be of the formX = , thus predicting anyons that interpolate bosons and fermions.B We note that, though the particle exchange is a dynamical process in the configuration-spaceB approach, the particle statistics itself is stille kinematical in the sense that it  is determined by the representation theory of 1 n . In one dimension, however, the situation is rather different: if R, the configuration space becomes a simply-connected convex set with boundary( ) (see section 2.1) such that any closed loops become homotopically equivalent. One might therefore think that multiparticleX = wavefunctions would not acquire any nontrivial phase under the particle exchange and there would arise only the trivial statistics (i.e., the Bose-Einstein statistics). This is, however, not the case because—as we will see in the path-integral formalism—identical particles still acquire a nontrivial phase every time multiparticle trajectorieshit the boundaries, just as in the case of a single particle on the half-line [15, 16]orinabox [17–19] (see also [20] for a textbook exposition). In addition, such a phase turns out to be a member of a one-dimensional unitary representation of the symmetric group n, thus reproducing the Bose-Fermi alternative again in one dimension. We will revisit these things in section 3. The rest of this section is devoted to a detailed analysis of theS configuration space for identical particles on R. We see that a coincidence point of particles corresponds to a codimension- boundary of the configuration space. We then discuss that two-body contact interactions aren generally described by the Robin boundary conditions at the codimension-1k + 1 boundaries. Since there are k such boundaries, it is possible to introduce distinct coupling constants in the -body problem of identical particles in one dimension. Furthermore, these coupling constants turn out to be ablen − to 1 depend on the coordinates. These are in starkn − contrast 1 to the boundary conditions (4n) and lead us to generalize the boson-fermion duality between the Lieb-Liniger and Cheon-Shigehara models. Finally, we discuss the boson-fermion mapping in our setup.

2.1 Configuration space of identical particles Let us begin with the definition of the -body configuration space in one dimension. As noted before, the configuration space of identical particles is given by first taking the Cartesian product of the one-particle configuration space, and thenn subtracting the coincidence points, and then identifying all the permuted points. The resultingn space in one dimension is the following orbit space:

 Rn n n (9) where Rn is the configuration space for distinguishable= ̊ /S , particles on R and given by

Rn Rn ̊ n n (10)

Here n is a set of coincidence points j ̊ k =where−Δ two. or more particles occupy the same point. In one dimension, such a set can be defined as the following vanishing locus of the Vandermonde Δ x = x 5 polynomial:

Rn n 1 ⋯ n j k (11) T 1≤j (9) can> x be identified Rn with the following bounded region in :n x > > x  Rn n 1 ⋯ n 1 ⋯ n (12)

This is the configuration space of =identical {(x , , x particles) ∈ ∶ inx one> dimension.> x }. Notice that this space is a convex set and hence simply connected4. Though it is not necessary forn deriving the boson-fermion duality, it may be instructive to point  out here that n can be factorized into the direct product of three distinct pieces—the space of the center-of-mass motion, the space of the hyperradial motion, and the space of the hyperangular motion. To see this, it is convenient to introduce the following normalized Jacobi coordinates:

1 ⋯ j j+1 j ⋯ (13a)

x1 + ⋯ + xn − jx n = t , j ∈ {1, , n − 1}, (13b) j(j + 1) x + + x We note that these are normalized = in√ the sense. that the coordinate transformation ⋯ ↦ n 1 n 1 ⋯ n is an transformation and hence preserves the dot product. Note also that n cor- responds to the center-of-mass coordinate and is invariant under n. It then follows( fromx , the, x ) defi- nitions( , , (13a) ) andSO (13b(n)) that there hold the identities x −x and x −x j−1 j+1 for 2 1 2 2j j−1 2j j ⋯ , from which one can show that the condition1 2 S j j+1 ⋯ is translated into √ 1 √2 u n u the condition ⋯ n(n−1) . Note that there is no= constraint on =, − meaning that+ the one- 1 2 n−1 n R  dimensionalj = {2, , n − space 1} n isu factored out from n. In order to seex > further x > factorizations,> x let us next introduce the hyperradius0 <  < < as follows:  (∋  ) 2 2 r 1 ⋯ n−1 2 t ⋅ n r =  + +  t 2 =  ⋅ −  1 ⋯ n w 1 = x x − (x + +2 x ) (14) n j k z 1≤j

4    It is easy to see that, if , then = × for×Ω any , . Thus is a convex set. Note that any convex set is simply connected. n n n x, y ∈ (1 − s)x + sy ∈ s ∈ [0, 1] 6 2 3

0 0 0

1 1

1 2 (a) Two-body relative space  . (b) Three-body relative space  . (c) Four-body relative space  .

Figure 1: Configuration spaces for1 the relative motion of two, three, and2 four identical particles in one dimension.3  R (a) 1 = { 1 ∈ ∶ 0 1 is just the half-line. The blue dot represents the codimension-1 boundary  R2 at which a two-body contact interaction takes place. (b) 2 1 2 1 2 is the infinite sector with the angle . The gray shaded region represents the impenetrable domain for the  <  } √ identical particles. The blue lines and the red dot represent the= {(codimension-1 ,  ) ∈ ∶ 0 and<  -2< boundaries3 } at  R3 which two- and three-body contact/3 interactions take place. (c) 3 1 2 3 1 2  3 is the infinite triangular pyramid. (To visualize 3, consider, e.g., the tetrakis hexahedron.) The gray shaded region represents the impenetrable domain. The blank white surfaces (including the√ blue √ = {( ,  ,  ) ∈ ∶0 <  < 3 < curves),6 } the red lines, and the green dot represent the codimension-1, -2, and -3 boundaries at which two-, three-, and four-body contact interactions take place.

R R where is the space of the center-of-mass motion, + the space of the hyperradial motion, and −2 the space of the hyperangular motion given by n n = { ∶ −∞ < < ∞} = {r ∶ r > 0} n ΩR −1 2 2 −2 1 ⋯ −1 1 ⋯ −1 1 ⋯ −1 (16) T w U n n n n(n − 1) n Ω = (̂ , , ̂ ) ∈ ∶ ̂ + + ̂n =1 & 0 < ̂ < < ̂ . We note that the last factor −2 in (15) must be discarded if . Note also2 that the subspace  R 5 −1 + −2 is nothing but the relativespace in [10] that describes the relative motion of identical n particles. Typical examples ofΩ the relative space are depicted in figuren = 21. n n Now,= as×Ω can be observed from figure 1, there are a number of nontrivial boundaries in the relative  space −1. For example, for (see figure 1c), there are(i) threecodimension-1boundaries, (ii) three codimension-2 boundaries, and (iii) a single codimension-3 boundary, which, in the original Cartesian n coordinates, correspond to (i)n = 41 2 3 4 , 1 2 3 4 , 1 2 3 4 , (ii) 1 2 3 4 , 1 2 3 4 , 1 2 3 4 , and (iii) 1 2 3 4 , respectively. In general, a coincidence point{ ofx =particlesx > x corresponds> x } {x to> one x = ofx the> codimension- x } {x > x > x boundaries= x }   {ofx = −1x (or= x >). x } {x = x > x = x } {x > x = x = x } {x = x = x = x } To summarize, we have seen thatk -body contact interactions take place at a codimension-(k − 1) boundaryn of then configuration space  . Since the purpose of the present paper is to derive and generalize the dual two-body contactk interactions for bosons and fermions, in what follows we(k − will 1) concentrate on only the codimension-1 boundaries.n

2.2 Two-body boundary conditions In order to construct a quantum theory on  , we have to specify boundary conditions for wavefunc- tions. Below we will do this by imposingthe probabilityconservation at the codimension-1 boundaries. Let us first note that a two-body contactn interaction between the th and th particles takes place at the following codimension- boundary: j (j + 1) 2-body R 1 ⋯ 1 ⋯ +1 ⋯ (17) n 5 n j j n The relative space can) alson,j be written= {(x as, , x ) ∈ ∶ x >R > x = x > > x }, . n−1 u n(n−1) n−1 1 n−1 1 2 n−1 = {( , ⋯ , 7 )∈ ∶0 <  < ⋯ <  } where ⋯ . As originally discussed by Leinaas and Myrheim [10], in order to ensure the probability conservation, the normal component of the probability current density must vanish at the boundary.j ∈ {1Thus, , we n − impose 1} the following condition:

n ⋅j on 2-body (18)

6 j 2-bodyn,j where n stands for a normal vector to the= boundary 0 ) ,and j is the -body probability current density defined by j ) n,j n j (19)

ℏ  Here 1 ⋯ is a multiparticle= wavefunction ( − (( on ) ., ⋯ is the -body dif- 2im  ferential operator, and the overline stands for the complex conjugate. Substituting) ) (19) into (18) we n n 1 n get = (x , , x ) ( = ( )x , , )x ) n

n ⋅ n ⋅ on 2-body (20)

Note that this is a quadratic equationj of .j However, it can in fact be linearized and enjoys a one- ( ( ) − ( ( ) = 0 ) n,j . parameter family of solutions. As is well-known, the solution to the equation (20) is given by the following Robin boundary condition:

n ⋅ on 2-body (21)

j 1 n,j where is a real parameter with the( dimension − j = of 0 length.) We emphasize, that may depend on the coordinates parallel to the boundary. Naively,a such coordinate dependence would break the translation j j invariance.a As originally noted in [10], this is true for . However, for ≥ ,a can depend on the coordinates without spoiling the translation invariance. We will revisit this possibility and resulting j scale- and translation-invariant two-body contact interactin = 2ons in section 4n. 3 a 2-body R Now, since the boundary is the codimension-1 surface +1 in , the normal vector can be simply written as follows: n j j ) n,j x − x = 0 n +1 ⋯ ⋯ (22) j j j = ((x − x ) from which we find that the Robin boundary condition (21) is cast into the following form: = (0, , 0, 1, −1, 0, , 0),

on 2-body (23) +1 ) ) 1 This is the boundary condition that− describes the− two-body = 0 conta) ctn,j interaction. for identical particles. 0 j j 1 j It should be noted that Leinaas)x and)x Myrheima interpreted the parameter as a statistics param- eter that interpolates the Bose-Einstein and Fermi-Dirac statistics [10], because continuously in- j terpolates the Neumann boundary condition (i.e., the boundary condition fora free bosons) and the j Dirichlet boundary condition (i.e., the boundary condition for free fermions). Theya then advocated the existence of intermediate statistics in one dimension. In the present paper, however, our take is different: the parameter just describes the two-body contact interaction rather than the statistics. As we will see in section 3, for bosons, the Robin boundary condition (23) is translated into the - j function potential whosea coupling constant is and whose support is the codimension-1 singularity R 1 ⋯ +1 ⋯ in . For fermions, on the other hand, (23) turns out to become the -function potential whose coupling constant is j and whose support is ⋯ ⋯ . n 1/a 1 +1 j j n {x 6 > > x = x > > x } ̊ Note that normal vectors become ill-defined at the codimensij on- ≥ boundaries, because thesej boundariesj are cornern "singularities in general; see figures 1b and 1c. In this work wea will not touchupon boundary{x conditions> > x at= thxese singularities.> > x } k( 2) 8 2.3 Boson-fermion mapping Before closing this section, let us discuss here the boson-fermion mapping in terms of multiparticle wavefunctions on R . To this end, let us first suppose that we find a normalized wavefunction on the region ⋯ (i.e., the configuration space  ). Let us then extend this wavefunction by 1 n introducing the following̊ two distinct wavefunctions on the region (1) ⋯ ( ): (x) n n x > > x   n B x > > x (24a) 1 F(x) ∶= √ (x), (24b) n! 1 As runs through all possible permutations, (x) ∶= √ eqs.sgn( (24a)) and (x) (24b. ) define normalized wavefunctions on R . By construction, it is obvious that B andn! F are totally symmetric and antisymmetric under the permutation of coordinates, thus providing the wavefunctions of identical spinless bosons and  n R fermions̊ on . It is also obvious by construction that there holds the identity F B on the region ⋯ . An alternative equivalent expression for this is the following identity n (1) ( ) on R : ̊ (x) = sgn() (x)   n n x > > x ̊ F B R (25) 1≤ ≤ n j k where here stands for(x) the = signÇ function,sgn(x − x ) (x)., This∀x is∈ the̊ , celebrated boson-fermion H j

3 Dual description of the Feynman kernel on  R

In the previous section, we have presented a detailed analysis of the configurationn space of identical n n particles in one dimension. Note, however, that the particle statistics is= still̊ unclear/S at this stage: it is not clear whether and how multiparticle wavefunctions on  acquire a phase under then process of particle exchange. As noted at the beginning of section 2, the particle exchange is a dynamical pro- cess in the configuration-space approach. Hence it is natural ton expect that particle-exchange phases may show up by studying dynamics. In general, dynamics in quantum mechanics is described by the Feynman kernel—the integral kernel of the time-evolution operator—which, as is well-known, can be studied by the path-integral formalism. In fact, the configuration-space approach and the path-integral formalism were known to be intimately connected. The key was the covering space approach to the path integral on multiply-connectedspaces, whichwas initiatedbySchulman[21] and later generalized by Laidlaw and DeWitt [9] and byDowker[22] (see also [23–25]). In this approach, one first constructs a multiply-connected space as , where is a (simply-connected) universal covering space of and is a discrete subgroup of the isometry of that acts freely on (without fixed points). Then, the path integral on is generallyQ Q = givenQž/Γ by a weightedQž sum of the path integrals on with weight QfactorsΓ given by a one-dimensional unitary representationQž of . All theQž weight factors are linked with homotopically distinctQ paths and—since is isomorphic to the fundamental group 1 —coincideQž with the particle-exchange phases in the configuration-space approach.Γ In this way, the path-integral ap- proach to the particle statistics was successfulΓ in deriving the Bose-Fermi alternative (Q) for ≥ [9] and the braid-group statistics for [26]. In one dimension, however, the situation is rather different again. This is because the configurationd 3 R space  is not a multiply-connectedd = 2 space and does not fit into the form . In fact, to the best of our knowledge, the path-integral derivation of the particle statistics in one dimension is n n n still missing.= ̊ /S Q = Q/Þ

9 In this section we study the Feynman kernel for identical particles in one dimension by using the path-integral formalism. We will see that—though R is not a universal covering space of  and is not the fundamental group of  either—the covering-space-like approach still works in n n one dimension: the Feynman kernel on  R turns̊ out to be given by a weighted sum of then Feynmann kernels on R with weight factorsn given by a one-dimensional unitary representation of .7 S n As we will see shortly, this leads to the Bose-Fermin n alternative as well as a generalized form of the n = ̊ /S n boson-fermion duality.̊ S To begin with, let us consider the simplest situation where the particles freely propagate in the bulk yet interact only through the two-body contact interactions described by (23). The dynamics of such a system is described by the following time-dependent Schrödinger equation on  :

2 2  n (26)

) ℏ n with the Robin boundary conditionsiℏ (+23). A( standard (x, t) approach = 0, ∀x to ∈ solve, this problem is to find out the 0 )t 2m 1 Feynman kernel  xp(− ) on  , which is the coordinate representation of the time-evolution operator and must satisfy the following properties: n i ℏ n • Property 1.K (Composition(x, y; t) = ⟨x law)| e Ht |y⟩

 z  ( z; 1)  (z y; 2) =  ( ; 1 + 2) ∀ ∈ (27) ‰ n n n n • Property 2. (Initialnd condition)K x, t K , t K x, y t t , x, y ,

 ( ; 0) = ( − ) ∀ ∈  (28)

n n • Property 3. (Unitarity) K x, y  x y , x, y ,

 ( ; ) =  ( ; − ) ∀ ∈  (29)

n n n • Property 4. (SchrödingerK equation)x, y t K y, x t , x, y , 2 + 2  ( ; ) = 0 ∀ ∈  (30) 2 ) ℏ n n iℏ (x K x, y t , x, y , • Property 5. (Two-body0 boundary)t m conditions)1 1 2-body −  ( ; ) −  ( ; ) = 0 ∀ ∈  ∀ ∈  (31) +1 n n ) ) n,j n j j K x, y t j K x, y t , x ) , y . Notice that the0 first)x four)x properties1 are justa the coordinate representations of the composition law † −1 = + , the initial condition 0 = 1, the unitarity = = − , and the Schrödinger equation for the time-evolution operator , where stands for the Hamiltonian ( 1 2− ) 1 =2 0 = exp(− ) t t t t t t t Uof theU) system.U Once we find such aU Feynman kernel, theU solutioni U toUthe time-dependent Schrödinger t t equationiℏ )t H (U26) can be written as the following integralU transformℏ Ht of the initH ial wavefunction at = 0:

( ) =  ( ; ) ( 0) ∀ ∈  t (32) ‰

n n 7If one wants to use covering-space x, t language,ndy oneK shouldx, y disct ardy the, , subtractionx procedure. and consider the orbifold  R [27]. In this case, one can say that the path integral on  is given by a weighted sum of the path integrals on R the orbifoldn universal cover and that weight factors are given by a one-dimensional unitary representation of the orbifold n n orb  n fundamental̆ = /S group n . (For orbifolds, see, e.g., the Thurston’s̆ lecture notes [28].) Notice that, in one dimension, the difference between  R ⋯ ≥ ⋯ ≥ and  R ⋯ ⋯ is merely 1 n n the boundary: the former ( ̆ includes) ≅ Sn the boundary but the latter does not. Hence itn is almost a of preference which one n n 1 Rn 1 n n n 1 n 1 n we should use. In this paper̆ = we will/S use= {(x , , x ) ∶ xin order tox make} the conceptual= ̊ /S = transition {(x , , x from) ∶ x > to> x≥} smooth. n n ̊ n = /S 10 d =1 d 2 It should be emphasized that, if the Feynman kernel satisfies (31), the wavefunction given by the inte- gral transform (32) automatically satisfies the Robin boundary conditions (23). R Now, as we will prove in appendix A, the Feynman kernel on  = / can be constructed in almost the same way as the Dowker’s covering space method [22] and written as follows: n n ̊ n  S  ( ; ) = É ( ) R ( ; ) ∀ ∈ (33) ∈ n n ̊ n where ∶ → (1) isK a one-dimensionalx, y t  Sn unitary K x representation, y t , x, y of . Here, R is the Feynman R kernel on and assumed to satisfy the following conditions: n n n S U S K ̊ • Assumptionn 1. (Composition law) ̊ R R 1 R 2 R 1 2 (34) ‰R z ( z; ) (z y; ) = ( ; + ) ∀ ∈ n n n n n ̊ ̊ ̊ • Assumption 2. (Initial̊ d K condition)x, t K , t K x, y t t , x, y ̊ , R R ( ; 0) = ( − ) ∀ ∈ (35)

n n • Assumption 3. (Unitarity) K ̊ x, y  x y , x, y ̊ , R R ( ; ) = R ( ; − ) ∀ ∈ (36)

n n n • Assumption 4. (SchrödingerK ̊ equation)x, y t K ̊ y, x t , x, y ̊ , 2 2 R + R ( ; ) = 0 ∀ ∈ (37) 2 n n ) ℏ x ̊ • Assumption 5. (Permutationiℏ invariance)( K x, y t , x, y ̊ , 0 )t m 1 R R ( ; ) = R ( ; ) ∀ ∈ ∀ ∈ (38)

n n n n with some connection conditionsK ̊ x, aty thet singularK ̊ x, locus y t Δ, inx order, y to̊ , glue the S!,disconnected regions together. It is these connection conditions that we wish to uncover below. Before doing this, however, n let us first present a brief derivation of the formula (33) by following the Dowker’sn argument [22]. R Let ( ) be an equivariant function on that satisfies ( ) = ( ) ( ) for any ∈ . Notice that, if [B] ( [F]), such an equivariant function is nothing but the wavefunction of = = n R n identical bosonsx, t (fermions) on . Then, the solution̊ to the time-dependent x, t  Schrödinger x, t equation Son R is given by the following integral transform of the initial wavefunction: n n ̊ ̊ R ( ) = ‰R ( ; ) ( 0) n n ̊ x, t ̊ dy K x, y t y, = R ( ; ) ( 0) ‰ É H ∈ I n n ̊ dy  Sn K x, y t y, = É R ( ; ) ( ) ( 0) ‰ H ∈ I n n ̊ dy  Sn K x, y t  y, R = ( ) R ( ; ) ( 0) ∀ ∈ (39) ‰ É H ∈ I n n ̊ n ̊ Here the second equality follows fromdy  theSn following  K x integral, y t form y,ula, (forx the proof,. see appendix B):

( ) = ( ) (40) ‰R ‰ É H ∈ I n n ̊ dy f y dy  Sn f y , 11 x3 x2 x1 x3 x2 x1 x3 x2 x1 x3 x2 x1 x3 x2 x1 x3 x2 x1

(a) identity. (b) . (c) . (d) . (e) . (f) . y3 y2 y1 y3 y2 y1 y3 y2 y1 y3 y2 y1 y3 y2 y1 y3 y2 y1

Figure 2: Typical trajectories contained1 in the weighted2 sum  1 2 2 1 . Any1 2 1 per-  =  =   =   =3  ∈3=   R3  =    mutation 3 can be decomposed into a product of the adjacent transpositions 1 and 2 . Correspondingly, any weight factor Kcan(x be, y; decomposedt) = ∑ S into()K å product(x, y; t of) 1 and 2 . The physical∈ S meaning of this is that the Feynman kernel acquires the particle-exchange = (1, 2) phase  =every (2, 3) time two adjacent particles collide with ( each) other in the course of the time-evolution. ( ) This is ( equivalent ) to the statement that the Feynman kernel acquires the particle-exchange phase every timej the three-particle trajectory hits the codimension-1 boundary 2-body. ( ) 3 (j) ) ,j where is an arbitrarytest function on R . By restricting the variable to  and then comparing (39) with (32), one arrives at the formula (33). A proof that eq. (33) indeed satisfies the conditions (27)–(30) n n is presentedf in appendix A. It is now obvious̊ from the above derivationx that the weight factor ( ) in (33) describes a particle-exchange phase. More precisely, ( ) is an accumulation of particle-exchange phases for two adjacent particles in the course of the time-evolution; see figure 2 for the case of = 3. Now, as noted repeatedly, there are just two distinct one-dimens ional unitary representations of [B] [F] : the totally symmetric representation and the totally antisymmetric representation .n This simple mathematical fact has two distinct physical meanings here. To see this, let us first suppose that n  S R is given. Then the formula (33) implies that there exist two distinct Feynman kernels on : n [B] [B] ̊ n(41a) K  ( ; ) = É ( ) R ( ; ) ∈ n [F]n [F] ̊ n (41b) K (x, y; t) = ÉS ()KR (x, y; t), ∈ n n ̊ Since R is normally constructedK fromx, y t classical Sn theory K viax the, y Feynman’t . s path-integral quantiza- tion, eqs. (41a) and (41b) indicate that there exist two inequivalent quantizations depending on the n [B/F] particle-exchangeK ̊ phases ( ). Hence the first meaning is about the particle statistics: in one di- mension, there just exists the Bose-Fermi alternative rather than the intermediate statistics of Leinaas and Myrheim [10]. To see another meaning, let us next suppose that  is given. Then the for- mula (33) implies that there may exist two distinct Feynman kernels on R satisfying the following n equalities: K n [B] [B] ̊  ( ; ) = É ( ) R ( ; ) ∈ n n [F] [F]̊ n (42) K x, y t = ÉS ()KR ( x, y; t) ∈ n ̊ Since  determines the dynamics as well as energySn  spectrumK x,  ofy identical-pt . article systems, eq. (42) indicates that the two distinct systems on R described by [B] and [F] are, if they exist, completely n R R isospectral.K Hence the second meaning is about the boson-fermion duality: there may exist bosonic n n n and fermionic systems whose energy spectrå are completelyK eqů ivalent.K ̊ Let us finally derive a generalization of the boson-fermion duality by using (42), which can be [B] [F] achieved by studying the connection conditions for R and R at the codimension-1 boundaries. To this end, let us first note that any element of can be classified into either even or odd permutations. n n In other words, the symmetric group can be decomposedK ̊ intoK ̊ the following coset decomposition: n S n (43) S = ∪ where is the alternating group of order thatn consistsn n of only even permutations. is an arbi- S A A , ∈ trary transposition and = { ∶ ∈ } is the right coset that consists of only odd permutations. n n A n  S n n A    A 12 Then, corresponding to (43), the formula (33) can be decomposed into the following form:

 ( ; ) = É ( ) R ( ; ) + ( ) R ( ; ) ∈ n n n ̊ −1 ̊ −1 −1 (44) K x, y t =  An ( ) K R (x, y t ; ) + ( K) R x( , y t  ; ) ∈ n n ̊ ̊ ÉAn  K  x, y t  K   x, y t  , −1 −1 where we have used ( ) = ( ) ( ) and R ( ; ) = R ( ; ) = R ( ; ) (∀ ∈ ) which follows from the permutation invariance (38). Now we choose as the adjacent transposition n n n ̊ ̊ ̊ n1 = = ( + 1) which  just swaps  and K+1. Then,x, y byt applyingK  thex differential, y t K operator x, y t −  −S to (44), we get  ) ) j j j j j+1 j   j,j x x )x )x a 1 −  ( ; ) −  ( ; ) +1 ) ) n n j j K x, y t j K −1 x, y t 0= )x ()x) 1 − Ra ( ; ) ∈ +1 n ) ) ̊ Én  j j K −1 −1x, y t − A ( ) 40)x − )x 1R ( ; ) +1 ) ) n 1 j −1 ̊ j −1 −1 −  R ( j ; j) +K ( ) R( x, y t ; ) 0)x )x 1 5 n n ̊ j ̊ j j K  x, y t  K   x, y t = Éa 0( ) − R (z y; ) 1 ∈ 4 ( ) ( +1) z= n ó ) ) ̊ ó −1  An   j  j K , t ó  x − ( ) 0)z − )z 1R (z y; ) ó ( ) ( +1) z= n j ) ) ̊ ó −1 −1 ó j 1   j  j K , t ó   x (45) − 0R)z(z y; ))z +1 ( ) R (z y;ó ) z= z= 15 n n ̊ ó −1 j ̊ ó −1 −1 ó ó j j K , t ó  x  K , t ó   x , a 0 ó ó where in the last equality we have introduced a new variable z = −1x and used the relation x = z (i.e., = ( ) for any ) in the first and third terms. Similarly, in the second and fourth terms of the −1 −1 last line we have introduced z = x and used the relation x =  z, which gives +1 = ( ( +1))= , k  k , and for . It should be noted that in ( ) x =z ( ( )) = k( +1) = ( ( )) = ( ) ∉ { + 1} j x j j j   j (45) is in the region 1 ⋯  , which means that z in the first and third termsx is in thez region  j j  j j  j k  j k  k z (1) x ⋯ z ( ) z ( +1) ⋯x z( ) whereasz z in thek secondj,j and fourth terms is in the region n (1) ⋯ ( +1) x( ) > ⋯ > x( ). Hence, in order for  to satisfy the Robin boundary condition (31 ) as →j , the j Feynman kernel n on R should satisfy the following connection condition at z > − >+1 z >0+ z > > z n the codimension-1 j singularity j  n ⋯ ⋯ : z > >z >z > >z{ (1) ( ) n= ( +1) K ( )} j j x x ̊   j  j  n z > >z z > >z ó ó 0 = − R (z y; )ó 0 ( ) ( +1) 1 ó − =0 n ) ) ̊ (j) (j+1)ó +  j  j K , t z z ó − )z( ) )z − R (z y; )ó 0 ( ) ( +1) 1 ó − =0 n j ) ) ̊ ó (j+1) (j)ó + 1   j ó  j K , t z z ó (46) − R)z(z y; )ó)z + ( ) R (z y; )ó 0 ó − =0 ó − =0 1 n n ̊ j ̊ j K , t (j) (j+1) +  K , t (j+1) (j) + , a z z z z where ∈ {1 ⋯ − 1} and ∈ . Now it is obvious that, in the totally symmetric representation

n j , , n  A 13 [B] [B] [B] = , where ( ) = 1, we have the following connection condition for R = R :

n j [B] ó ̊ n ó ̊ 0 = − R (x y; )ó K K 0 ( ) ( +1) 1 ó − =0 ) ) n ̊ [B] ó(j) (j+1) +  j  j K , t x ó x − )x )x− R (x y; )ó 0 ( ) ( +1) 1 ó − =0 n ) ) ̊ [B] ó K , t [B](j) (j+1) ó− 1  j  j ó x x ó (47) − )x R (x)xy; )ó + R (x y; )ó H ó − =0 ó − =0 I n n ̊ ̊ [F] where we have renamed z toj xK. In contrast,, t x in(j) thex(j+1) totally+ K antisymmetric, t x(j) representationx(j+1) − , = , where [F] a [F] ( ) = sgn( ) = −1, we have the following connection condition for R = R :

n j j [F] ó ̊ n ó ̊   0 = − R (x y; )ó K K 0 ( ) ( +1) 1 ó − =0 ) ) n ̊ [F] ó(j) (j+1) +  j  j K , t x ó x + )x )x− R (x y; )ó 0 ( ) ( +1) 1 ó − =0 n ) ) ̊ [F] ó K , t [F](j) (j+1) ó − 1  j  j ó x x ó (48) − )x R (x)xy; )ó − R (x y; )ó H ó − =0 ó − =0 I n n ̊ ̊ Now, these connectionj conditionsK , aret nothingx(j) x(j+1) but+ (aK part of), thex connection(j) x(j+1) − conditions. for the - and -function potentials.a In fact, it follows from these connection conditions that the wavefunctions constructed from the integral transform (39) satisfy the conditions (2a)or(2b) with being replaced by ". Note that the continuity conditions for the wavefunction (2b) and its derivative (3b) are not necessary here because the totally symmetric function and the derivative of totallya antisymmetric j functiona automatically becomes continuous at the coincidence points. Thus, reversing the argument used to arrive at (2a)–(3b) from (1a) and (1b), we obtain the following dual Hamiltonians for identical bosons and fermions: 2 2 n B = − + B(x) (49a) 2 2 ℏ 2 HF = − ( + VF(x), (49b) 2m where ℏ H ( V , 2 −1 m 1 B(x) = É É Ç ( ( ) − ( +1)) ( ( ) − ( +1); ) (50a) n =1 ∈ L ∈{1 ⋯ −1}⧵{ } M ℏ  k  k  j  j aj 2 −1 n V j  A k , ,n j  x x  x x , F(x) = m É É Ç ( ( ) − ( +1)) ( ( ) − ( +1); ) (50b) n =1 ∈ L ∈{1 −1} { } M ℏ  k  k  j  j j Note that the factorV n ,x wherex is the step" x function,x isa introduced. in order ∏ ∈{1m j −1} {A} (k (,⋯) −,n (⧵ +1)j ) to guarantee the ordering (1) ⋯ ( ) = ( +1) ⋯ ( ). Note also that the total number of the codimension-1 singularitiesk ,⋯,n in R⧵ j is (−1)k  k ! , where stands for  x 2 ×x ( − 1)! = ( − 1) × 2 = ( − 1) × | | | | the order of the alternating group . Hence j  thej summation n −1 is indeed summing over all x >n >n xn x > > x ∑n =1 ∑ ∈ n n the codimension-1 singularities.̊ Finally, it is easyn to seen that then potentialn energiesA (50a)A and (50b) are n n invariant under and satisfy the identitiesA B/F( x) = B/F(x) forj any A ∈ . To summarize, by imposing the Robin boundary conditions to the Feynman kernel (42), we have [B] n [F] n found that R andS R , respectively, must satisfyV  the connectionV conditions S for the - and -function R potentials at then codimension-1n singularities in . The -body Hamiltonians that realize these con- nectionconditionsK ̊ forK ̊ identical bosons and fermions are givenby(49a) and (49b), bothof which" possess n − 1 distinct (coordinate-dependent) coupling constants.̊ n By construction, these models enjoy (i) the spectral equivalence, (ii) the boson-fermion mapping, and (iii) the strong-weak duality. n 14 4 Summary and discussion

In the present paper, we have revisited the boson-fermion duality in one dimension by using the configuration-space approach and the path-integral formalism. In section 1, we have first presented the detailed analysis of the configuration space for identical particles on R. We have shown that the two-body contact interactions for identical particles are generally described by the −1 distinct Robin boundary conditions (23), where the boundary-conditionn parameters ( = 1 ⋯ −1) may depend on R the coordinates parallel to the codimension-1n boundaries of  = / . In sectionn 2, we have then studied the dynamics of identical particles on R by using the Feynmanj kernel. We have first shown an j , , n n n that the Feynman kernel on  is generally given by (33)—the weighted̊ S sum of the Feynman kernels R on , where the weightn factor ( ) is a member of either the totally symmetric or totally antisym- metric representations of . Thisn result has two distinct physical meanings: the first is the existence n of the̊ Bose-Fermi alternative in one dimension, which is expressed by the equations (41a) and (41b), n and the second is the existenceS of the boson-fermion duality in one dimension, which is expressed by the identity (42). Then, by using (42), we have shown that—in order for the Feynman kernel on  to satisfy the Robin boundary conditions—the Feynman kernel on R must satisfy the connection condi- tions for the -function ( -function) potential if is the totally (anti)symmetric representation. Thisn n proves the boson-fermion duality between the systems describe̊ d by the -boson Hamiltonian (49a) and the -fermion Hamiltonian" (49b), which are natural generalizations of the Lieb-Liniger model (1a) and the Cheon-Shigehara model (1b), respectively. n Beforen closing the paper, let us finally comment on an implication for the possible coordinate de- pendence on . Asnotedinsection 2.2, maydepend on the coordinatesparallel to the codimension-1 boundaries. This opens up a possibility to realize scale-invariant two-body contact interactions which j j do not containa any dimensionful parameters.a A typical example for such coordinate dependence is given by

= ∈ {1 ⋯ − 1} (51) where are dimensionless reals and j is thej hyperradius defined in (14). Note that the hyperradius a g r, j , , n , (14) does not contain the center-of-mass coordinate and is invariant under the spatial translation j R ↦ g + for any ∈ . Hence inr this case the Hamiltonians (49a) and (49b) are also invariant n under the spatial translation such that the center-of-mass momenta are conserved. Note also that, j j 2-body xfor ≥x 3,c the hyperradiusc is nonvanishing at the codimension-1 boundary  . In fact, vanishes only at the codimension-( − 1) boundary { 1 = 2 = ⋯ = }. Therefore, in the three- or more-bodyn problems of identicalr particles, we can construct scale-invariant models) withoutn,j spoilingr n the translation invariance. Note, however,n that this continx uousx scale-invariancex could be broken down to a discrete scale-invariance in exactly the same way as the Efimov effect [29]. We will address this issue elsewhere.

A Proof of the path-integral formula (33)

Following the ideas presented in [30, 31], in this section we show that the formula (33) satisfies the properties (27)–(30) if is a one-dimensional unitary representation of and if R fulfills the condi- tions (34)–(38). Below we prove these four properties separately. n n S K ̊ Property 1. (Composition law) Let us first prove the composition law (27). By substituting (33) into the left-hand side of (27) we have

z  (x z; 1)  (z y; 2) ‰ n n nd K , t K¨ , t ¨ = É É ( ) ( ) ‰ z R (x z; 1) R (z y; 2) ∈ ∈ n n ¨ ̊ ̊ n n   z1>⋯>znd K ,  t K ,  t  S  S 15 ¨ ¨ = É É ( ) ‰ z R (x z; 1) R ( z y; 2) ∈ ∈ n n ¨ ̊ ̊ n n  ¨ z1>⋯>znd K ,  t K  ,  t ¨ = ÉS ÉS ( ) ‰ w R (x w; 1) R (w y; 2) ∈ ∈ n n ¨ −1 −1 ̊ ̊ n n ¨¨  w (1)>⋯>w (nd) K , t K , ¨¨ t = ÉS  (S ) É ‰ w R (x w; 1) R (w y; 2) ∈ ∈ n n ¨¨ ̊ ̊ n ¨¨ n −1(1) −1(n) ¨¨  S   S w R >⋯>w d RK , t K ,  t = É ( ) ‰R w (x w; 1) (w y; 2) ∈ n n ¨¨ ¨¨ n ¨¨̊ ̊ n ̊ = ÉS ( ) R d(x K y; 1, + t2) K ,  t ∈ n ¨¨ ̊ (52) =  Sn( x y; 1K+ 2) ,  t t where in the second equalityn we have used the assumptions that is a representation that satisfies K , t t , ( ) ( ¨) = ( ¨) and R satisfies the permutation invariance (38). The third equality follows from the change of the integration variable from to , the fourth equality the change of the n z w = z summation  variables  fromK ̊ and ¨ to and ¨¨ ∶= ¨, the fifth equality the fact that the region ⋯ R (1) ( ) covers the whole as runs through all possible permutations, and the sixth equality the assumption (34). Hence we have shown that (33) satisfies the composition law (27) if isa −1 −1   n     n representationw > > w of and if R satisfies the̊ permutation invariance (38) and the composition law (34).

n n Property 2. (InitialS condition)K ̊ Let us next prove the initial condition (28). By substituting (33) into the left-hand side of (28) we have

 (x y; 0) = É ( ) R (x y; 0) ∈ n n ̊ K , = ÉSn ()K(x − , y) ∈

=  (Sn) (x − y)  = (x − y) (53) e  e where the second equality follows from the assumption (35). In the third equality we have used the  , fact that, if is not the identity element , x − y cannot be zero for any x y ∈  , which leads to (x − y) = 0 for ≠ . The last equality follows from ( ) = 1 and y = y. Hence we have shown n that (33) satisfies the initial condition (28e) if is a representation of and, if R satisfies the initial condition (35).    e e e n n S K ̊ Property 3. (Unitarity) Let us next prove the unitarity (29). By substituting (33) into the left-hand side of (29) we have

 (x y; ) = É ( ) R (x y; ) ∈ n n −1 ̊ K , t = ÉSn () K R (, y xt; − ) ∈ n −1 ̊ −1 = ÉSn ( )KR (y , x;t − ) ∈ n  ̊ (54) =  Sn (yx; −K ) ,  t where thesecond equalityfollows from n and the assumption (36), thethirdequality −1 ( ) ∈K (1) , t , ( ) = ( −1) and the assumption (38). Hence we have shown that (33) satisfies the unitarity (29) if is a one-dimensional unitary representation  of U and if R satisfies the unitarity (36) as well  as the permutation invariance (38).  n n S K ̊ 16 Property 4. (Schrödinger equation) Let us finally prove that (33) satisfies the Schrödinger equation (30). By substituting (33) into the left-hand side of (30) we have

2 2 2 2 +  (x y; ) = É ( ) + R (x y; ) 0 2 1 ∈ 0 2 1 n n ) ℏ x ) ℏ x ̊ iℏ ( K , t = 0 n  iℏ ( K ,  t (55) )t m  S )t m where we have used the assumption (37). Hence we have shown that (33) satisfies the Schrödinger , equation (30) if R satisfies the Schrödinger equation (37).

n Now, as discussed̊ in section 3, eq. (33) also satisfies the Robin boundary conditions (31) if K R satisfies the connection conditions (46). Thus the formula (33) fulfills all the required properties (27)– n ̊ (31) if is a one-dimensional unitary representation of and if R satisfies the assumptions (34)–(K38) as well as the connection conditions (46). n n S K ̊ B Proof of the integral formula (40)

In this section we prove the integral formula (40). To this end, let be an arbitrary test function on R . Then we have n f ̊ ‰R y (y) = É ‰ y (y) ∈ n ̊ d f n y(1)>⋯>y(nd) −1f = ÉS ‰ z ( z) ∈

n 1 nd f   S z >⋯>z −1 (56) = ‰ z É ( z) H ∈ I

1 nd n f  , where in the second line we have changed thez >⋯ integration>z  S variable as y = −1z. By changing the notations as −1 → and z → y in the last line, we arrive at the formula (40).  References 

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