Coherent-state path integrals in the continuum: The SU(2) case
aa r X i v : . [ c ond - m a t . o t h e r] M a y Coherent-state path integrals in the continuum:The SU (2) case G. Kordas ∗ , D. Kalantzis , and A. I. Karanikas University of Athens, Physics Department,Panepistimiopolis, Ilissia15771 Athens, Greece
May 24, 2016
Abstract
We define the time-continuous spin coherent-state path integralin a way that is free from inconsistencies. The proposed definitionis used to reproduce known exact results. Such a formalism opensnew possibilities for applying approximations with improved accuracyand can be proven useful in a great variety of problems where spinHamiltonians are used.
The Feynman path integral formalism [1, 2] provides the most powerful toolfor taking into account quantum behavior via classical computations. Ide-ally suited for semi-classical calculations the path integral machinery helpshandling and understanding quantum mechanics, quantum field theories orstatistical physics [3].The extension of path integration into the ordinary complex plane C ,through the Glauber coherent-states [4], and in the complex compact non-flat manifold ¯ C , through the SU (2) spin coherent-states [5,6], has expandedits range of applications in many areas of physics and chemistry [3, 7]. Thedetails of these extensions and their utility for semi-classical approximationshave been discussed in a lot of excellent papers [8–14]. In almost all of themthe authors point out the fact that path integrals in the continuum are formallimiting expressions of an underlying discrete definition meaning that, in caseof discrepancies, one must refer to the discrete version of the integrals. Forexample, the spin-coherent state path integrals were thought to be unreliablein their continuous form and trustable only in their discretized form [15–20]. ∗ [email protected]
1t was only after the emergence in the continuum [12, 14] of the Solari’s [10]“extra phase” that the try for a trustable continuous formulation of thespin-coherent state path integration has been renewed.Nevertheless however, the problems still persist as inconsistencies andwrong results have been reported recently [21] even for simple and exactlysolvable systems when examined via coherent-state path integrals in thecontinuum. Interesting enough, even with the inclusion of the “extra phase”contribution, a system described by a Hamiltonian of the form ˆ H ∼ ˆ S z cannot be described correctly in terms of path integration in the continuum.However, after the extensive use of the continuous formulation in almostall the fields of quantum theory [3] or after the Berezin’s quantization schemefor non-flat manifolds like ¯ C [6], it would be at least awkward if it was tobe concluded that, in the framework of the coherent state path integrals, itis impossible to define a classical continuous action in correspondence witha quantum system.In a recent paper [22] we examined the case of the continuous formulationof path integrals in terms of the Glauber coherent states. We found that theinconsistencies disappear if one follows a certain recipe to define the classicalHamiltonian entering in the continuous action that weighs the paths in thecomplex plane. In the present work we extend our work undertaking thetask of establishing a connection between the quantum Hamiltonian andthe continuous action appropriate for path integration in the spin coherent-states basis and we discuss some aspects of the corresponding time sliceddefinition. In the context of the proposed formulation the path integrationcan be performed directly in the continuum without facing inconsistenciesand reproduces the exact results.The paper is organized as follows. In Sec. 2 we present our proposal inthe context of the simplest possible system ˆ H = ω ˆ S z which can be exactly(and correctly) handled by a lot of means. We examine this system inthe framework of the “standard” spin coherent state path integration andwe compare the result with the analysis based on our proposal. In Sec. 3we consider the case of a system described by a Hamiltonian of the formˆ H = ω ˆ S z for the description of which the standard approach breaks down.We prove that in the framework of our proposal the spin coherent-state pathintegration yields the correct answer. In the last section we summarize ourfindings and we comment on possible applications. To present our arguments we begin by considering the simplest possiblecase: a single particle with spin s in an external constant magnetic field.The Hamiltonian of such a system is ˆ H = ω ˆ S z and the representation of2he time evolution operator in the spin coherent state basis reads: G ( ζ ∗ b , ζ a ) = h ζ b | e − iT ˆ H | ζ a i . (1)The states of this basis can be defined [23] through the relation | ζ i = 1(1 + | ζ | ) s e ζ ˆ S − | s, s i = 1(1 + | ζ | ) s s X j = − s (cid:20) (2 s )!( s − j )!( s + j )! (cid:21) / ζ s − j | s, j i , (2)where | s, s i is the eigenstate of ˆ S z with the largest eigenvalue. The states(2) form an overcomplete basis on the compact non-flat manifold ¯ C , theone-point compactified complex plane, that is identified with the SU (2)homogeneous space, SU (2) /U (1) [11, 12]:ˆ I = 2 s + 1 π Z d ζ (1 + | ζ | ) , (3) Z d ζ ≡ Z d ℜ ζ Z d ℑ ζ. (4)This resolution of the identity can be used to present matrix elements likethe amplitude (1) as integrals over paths in the complex space ¯ C [11, 12]: G ( ζ ∗ b , ζ a ) = Z D µ ( ζ ) ζ ∗ ( T )= ζ ∗ a ζ (0)= ζ a e γ ba ( ζ ∗ ,ζ )+ iS [ ζ ∗ ,ζ ] . (5)In this expression Z D µ ( ζ ) ( · · · ) ≡ lim N →∞ N − Y n =1 s + 1 π Z d ζ n (1 + | ζ | ) ! ( · · · ) , (6) γ ba = s ln (1 + ζ ∗ b ζ ( T ))(1 + ζ a ζ ∗ (0))(1 + | ζ b | ) (1 + | ζ a | ) (7)and S = Z T dτ is ˙ ζζ ∗ − ˙ ζ ∗ ζ | ζ | − H ( ζ ∗ , ζ ) ! . (8)The classical Hamiltonian in the action (8) is supposed to have the following“standard” form H ( ζ ∗ , ζ ) = h ζ | ˆ H | ζ i . (9)For the case in hand one easily finds that H = ωs (cid:18) − | ζ | | ζ | (cid:19) . (10)3hus G ( ζ ∗ b , ζ a ) = Z D µ ( ζ ) ζ ∗ ( T )= ζ ∗ a ζ (0)= ζ a e γ ba ( ζ ∗ ,ζ )+ i R T dτ (cid:18) is ˙ ζζ ∗− ˙ ζ ∗ ζ | ζ | +2 ωs | ζ | | ζ | (cid:19) . (11)The functional integral appearing in the last equation can be exactly eval-uated directly in the continuum [11, 12, 14]. The procedure goes as fol-lows. Firstly, one finds the “classical” solutions pertaining to the action(8): ζ c ( τ ) = ζ a e iωτ , ζ ∗ c ( τ ) = ζ ∗ b e iω ( T − τ ) . Then, the change of variables ζ = ζ c + δζ, ζ ∗ = ζ ∗ c + δζ ∗ leads to a prefactor together with a deter-minant that encapsulates quantum fluctuations and which can be written interms of a functional integral over complex variables η : G ( ζ ∗ b , ζ a ) = e − iωT s (1 + ζ ∗ b ζ a e iωT ) s (1 + | ζ b | ) s (1 + | ζ a | ) s ×× Z D η η ∗ ( T )=0 η (0)=0 e i R T dτ ( i ( ˙ ηη ∗ − ˙ η ∗ η )+ ω | η | ) . (12)The fluctuating integral must be carefully evaluated because the resultstrongly depends [3] on the underlying discrete prescription that definesthe continuum version appearing in Eq. (12). To be concrete, let’ s considerthe following discrete version of this integral: I = lim N →∞ N − Y n =1 Z d η n π ! × (13) × exp ( N − X n =0 "
12 [( η ∗ n +1 − η ∗ n ) η n − ( η n +1 − η n ) η ∗ n +1 ] + iεωη ∗ n +1 η n , with ε = T /N . This integral can be straightforwardly evaluated [3] and theresult is I = 1.Roughly speaking , this result is connected with the possibility to makea change of variables η → ηe iωτ , η ∗ → η ∗ e − iωτ in the continuum level or η n → η n (1 + iωεn ) , η ∗ n → η ∗ n (1 − iωεn ) in the discrete level. This changeleaves intact the measure of the integration but not the action (in this sense itconstitutes an “anomaly”) [14] as it cancels the term ω | η | which has beentaken to be the continuum limit of the term εωη ∗ n +1 η n in the discretizedversion of the action.Consider now the following “symmetrized” definition of the fluctuating4ath integral: I S = lim N →∞ N − Y n =1 Z d η n π ! × (14) × exp ( N − X n =0 "
12 [( η ∗ n +1 − η ∗ n ) η n − ( η n +1 − η n ) η ∗ n +1 ] + iεωη ∗ n η n , where ε = T /N .Despite the fact that the continuum limit of the two expres-sions is the same the result of the calculation is different [3]: I S = e iωT/ .In the formal level, one can check that the previously discussed change ofvariables does not cancel the quadratic term in the symmetrized action.Leaving for later the discussion about which of the mathematically possibleapproaches is the physically correct one, we shall adopt here the version(13). In this case we find: G ( ζ ∗ b , ζ a ) = e − iωT s (1 + ζ ∗ b ζ a e iωT ) s (1 + | ζ b | ) s (1 + | ζ a | ) s . (15)By taking the trace of the amplitude (15) we can immediately confirm thatthe correct result is produced:2 s + 1 π Z d ζ (1 + | ζ | ) ) G ( ζ ∗ , ζ ) = e − iωT s s X p =0 e iωT p = s X j = − s e − iωT j = Tr { e − iT ˆ H } . (16)A warning for a possible pitfall in the above described procedure, and es-pecially in the form of the classical Hamiltonian, comes up [21] when onestarts to consider more complicated systems. As we shall discuss in the nextsection, the case of the simple but less trivial Hamiltonian ˆ H = ω ˆ S z cannotbe correctly analyzed through this formulation.Our next step is, now, to consider the same system at the limit s → ∞ .At this limit the spin coherent states (2) reduce [12, 23] to the harmonicoscillator coherent states: | ζ i −−−→ s →∞ e − | z | ∞ X n =0 z n √ n ! | i , ( ζ → z/ √ s ) . (17)At the same limit the functional integral reduces to a functional integralover the states (17): G ( ζ ∗ b , ζ a ) → G ( z ∗ b , z a ) = Z D z z ∗ ( T )= z ∗ b z (0)= z a e Γ ba + iS [ z ∗ ,z ] . (18)5ere Z D z ( · · · ) ≡ lim N →∞ N − Y n =1 Z d z n π ! ( · · · ) , (19)Γ ba = − (cid:0) | z b | + | z a | (cid:1) + 12 ( z ∗ b z ( T ) + z a z ∗ (0)) (20)and S = Z T dτ (cid:18) i zz ∗ − ˙ z ∗ z ) − H ( z ∗ , z ) (cid:19) . (21)As we have discussed in [22], the identification of the classical Hamiltonian inthis continuous expression is not a trivial task: In order to correctly performcalculations in the continuum, one must follow a certain route to define theclassical action that enters in the path integral. To describe how this canbe realized in the present case, we express the quantum Hamiltonian, ˆ H ,of our system in terms of the harmonic oscillator creation and annihilationoperators by making use of the Holstein-Primakoff transformation [24]:ˆ S z = s − ˆ α † ˆ α, ˆ S + = p s − ˆ α † ˆ α ˆ α, ˆ S − = ˆ α † p s − ˆ α † ˆ α. (22)Using as starting point the quantum Hamiltonian, ˆ H = H ( ˆ α † , ˆ α ) our pro-posal proceeds with the quadratures ˆ q = ( ˆ α † + ˆ α ) / √ , ˆ q = i ( ˆ α † − ˆ α ) / √ H F = H F (ˆ p, ˆ q ). Then, the classicalHamiltonian that must weigh the paths in the complex space spanned bythe coherent states {| z i} , is obtained by taking the classical counterpart ofˆ H F : ˆ H F → H F ( p, q ) . (23)In this expression the classical “momentum” and “position” variables aredefined through the following canonical transformation: p ≡ h z | ˆ p | z i = i √ z ∗ − z ) , (24) q ≡ h z | ˆ q | z i = 1 √ z ∗ + z ) . (25)For the case in hand we haveˆ H = ω ( s − ˆ α † ˆ α ) → ˆ H F = ω (cid:20) s −
12 (ˆ p + ˆ q ) + 12 (cid:21) →→ H F = ω (cid:20) s −
12 ( p + q ) + 12 (cid:21) = ω (cid:18) s − | z | + 12 (cid:19) . (26)6n this way the correlator (18) assumes the form: G F ( z ∗ b , z a ) = Z D z z ∗ ( T )= z ∗ b z (0)= z a e Γ ba + i R T dτ ( i ( ˙ zz ∗ − ˙ z ∗ z ) − H F ) (27)= e − iωT s − i ωT Z D z z ∗ ( T )= z ∗ b z (0)= z a e Γ ba + i R T dτ ( i ( ˙ zz ∗ − ˙ z ∗ z )+ ω | z | ) . The change of variables z ( τ ) = z a e iωτ + η ( τ ) , (28) z ∗ ( τ ) = z ∗ b e iω ( T − τ ) + η ∗ ( τ ) (29)reduces the amplitude (27) into the form: G F ( z ∗ b , za ) = e − iωT s − i ωT e − ( | z b | + | z a | )+ z ∗ b z a e iωT ×× Z D η η ∗ ( T )=0 η (0)=0 e i R T dτ ( i ( ˙ ηη ∗ − ˙ η ∗ η )+ ω | η | ) . (30)The functional integral in the last equation is the same as the one appearingin Eq.(12) but now we have the necessary information to decide which oneof the discretized definitions (13) or (14) is physically correct [3]. Sincethe classical action in the functional integrals (27) or (30) arose from theoscillator action by means of the canonical transformations (24), (25) andsince the classical dynamics are invariant under canonical transformations,the associated amplitudes must be the same: Z D η η ∗ ( T )=0 η (0)=0 e i R T dτ ( i ( ˙ ηη ∗ − ˙ η ∗ η )+ ω | η | ) = h | e i T ω (ˆ p +ˆ q ) | i = h | e iT ω ˆ α † ˆ α | i e i T ω = e i T ω . (31)This result fixes the physically correct time-slicing to the symmetric one (seeEq.(14)). Consequently G F ( z ∗ b , z a ) = e iωT s e − ( | z b | + | z a | )+ z ∗ b z a e iωT . (32)By taking the trace of this expression we arrive at the correct result:Tr( e − iT ˆ H ) s →∞ = Tr G F = e − iωT s s X p =0 e iωT p = ∞ X j = −∞ e − iωT j . (33)7t is obvious now that this analysis raises a question about the validity of theresult (15): Suppose that we take the limit s → ∞ of the integral appearingin Eq. (11) Z D µ ( ζ ) ζ ∗ ( T )= ζ ∗ b ζ (0)= ζ a e γ ba ( ζ ∗ ,ζ )+ i R T dτ (cid:18) is ˙ ζζ ∗− ˙ ζ ∗ ζ | ζ | +2 sω | ζ | | ζ | (cid:19) −−−→ s →∞ Z D z z ∗ ( T )= z ∗ b z (0)= z a e Γ ba + i R T dτ ( i ( ˙ zz ∗ − ˙ z ∗ z )+ ω | z | ) = h z b | e i ωT (ˆ p +ˆ q ) | z a i . (34)The last amplitude can be easily calculated in configuration space: h z b | e i ωT (ˆ p +ˆ q ) | z a i = Z dx Z dx ′ h z b | x ih x | e i ωT (ˆ p +ˆ q ) | x ′ ih x ′ | z a i = e − ( | z b | + | z a | )+ z ∗ b z a e iωT + i ωT . (35)Obviously the limit s → ∞ of the result (12) does not coincide, as it should,with the last conclusion except if the symmetrized version is adopted. Onthe other hand , if we adopt this version the calculation of the correlator (5)yields the result G ′ ( ζ ∗ b , ζ a ) = e − iωT s (1 + ζ ∗ b ζ a e iωT ) s (1 + | ζ b | ) s (1 + | ζ a | ) s e i ωT (36)which is incorrect Tr { G ′ } = e i ωT s X j = − s e − iωT j . (37)To resolve the puzzle we ought to assume that the procedure described inEq. (23), for the identification of the classical continuum Hamiltonian, doesnot pertain to the asymptotic limit only but it is valid for all spin values.What distinguishes each case is the canonical transformation that definesthe classical “momentum” and “position” variables.Thus, for the realization of the recipe (23) for finite s , and in accordancewith Eqs. (24) and (25), we are led to define the classical variables q ≡ h ζ | ˆ α † + ˆ α √ | ζ i , (38) p ≡ i h ζ | ˆ α † − ˆ α √ | ζ i . (39)Using the Holstein-Primakoff transformation and the fact that h ζ | ˆ S − | ζ i = 2 s | ζ | ζ ∗ , (40) h ζ | ˆ S + | ζ i = 2 s | ζ | ζ, (41)8e immediately determine the classical variables (38) and (39) to have theform: q = 1 √ ζ ∗ + ζ ) s s | ζ | , (42) p = i √ ζ ∗ − ζ ) s s | ζ | . (43)Note that at the asymptotic limit s → ∞ the variables (24) and (25) arerecovered.Thus, according to our prescription, the Hamiltonian entering in theintegral (5) is not the “standard” one as indicated in Eq. (10) but the oneproduced after the canonical change of variables (42) and (43) that definesthe Feynman version of (10): H F = ω (cid:20) s −
12 ( p + q ) + 12 (cid:21) = ω (cid:18) s − s | ζ | | ζ | + 12 (cid:19) . (44)Repeating now the steps that led from Eq. (11) to Eq. (12) we find G ′ ( ζ ∗ b , ζ a ) = e − iωT s (1 + ζ ∗ b ζ a e iωT ) s e i ωT (1 + | ζ b | ) s (1 + | ζ a | ) s ×× Z D η η ∗ ( T )=0 η (0)=0 e i R T dτ ( i ( ˙ ηη ∗ − ˙ η ∗ η )+ ω | η | ) . (45)Adopting the result (30) we get the correct answer.The main conclusion of this section is that the use of the Feynman Hamil-tonian instead of the standard one together with the symmetrized definitionof the path integrals leads, without inconsistencies, to the correct result.However, the example we used was very simple and we have to con-sider more complicated Hamiltonians in order to check our proposal. Inwhat follows we shall call “standard” the procedure that adopts the form H ( ζ ∗ , ζ ) = h ζ | ˆ H | ζ i for the classical Hamiltonian and takes into considerationpresence of the “extra phase” factor. In this section we shall consider the less trivial Hamiltonian ˆ H = ω ˆ S z andthe correlator G ( ζ ∗ b , ζ a ) = h ζ b | e − iT ˆ H | ζ a i . (46)Due to the simplicity of the system this amplitude can be exactly evaluated: G ( ζ ∗ b , ζ a ) = s X j = − s e − iωT j h ζ b | s, j ih s, j | ζ a i , (47)9ith h ζ | s, j i = 1(1 + | ζ | ) s (cid:20) (2 s )!( s − j )!( s + j )! (cid:21) / ζ s − j . (48)However, the attempt to get the result (47) using the standard rules forperforming coherent- state path integration in the continuum, fails. Aswe shall confirm the calculation of Tr n e − iT ˆ H o in the standard framework(with the inclusion of the extra phase) yields the result [21]Tr n e − iT ˆ H o = s X j = − s e − iωT j + iωT j − s s , (49)which is wrong for every finite s except for s = 1 /
2. The correct answer isrecovered only at the asymptotic (classical) limit s → ∞ .However, the calculation in the continuum can be successfully executedby following the proposal presented in the previous section the main ingredi-ent of which is the identification of the classical Hamiltonian entering in thepath integral representation of the amplitude (46). To this end we appeal,once again, to the Holstein-Primakoff transformation to writeˆ H = ω (cid:16) s − s ˆ α † ˆ α + ˆ α † ˆ α ˆ α † ˆ α (cid:17) . (50)Expressing the annihilation and creation operators in terms of the quadra-tures we findˆ H F = ω " s − s (cid:18) ˆ p + ˆ q − (cid:19) + (cid:18) ˆ p + ˆ q − (cid:19) . (51)According to our prescription to obtain the classical Hamiltonian we replacethe “position” and “momentum” operators by their classical values (42) and(43) H F = ω " s − s (cid:18) p + q − (cid:19) + (cid:18) p + q − (cid:19) (52)= ω (cid:18) s + s + 14 (cid:19) + ω (2 s ) (cid:18) | ζ | | ζ | (cid:19) − ω s (2 s + 1) | ζ | | ζ | . It would be helpful at this point to write the form of the classical Hamilto-nian had we adopted the “standard” prescription H = h ζ | ˆ H | ζ i = ωs + ω s (2 s − (cid:18) | ζ | | ζ | (cid:19) − ω s (2 s − | ζ | | ζ | . (53)Obviously the two expressions are the same only at the asymptotic limit s → ∞ . In the framework of our proposal the amplitude (46) assumes the10orm G F ( ζ ∗ b , ζ a ) = Z D µ ( ζ ) ζ ∗ ( T )= ζ ∗ b ζ (0)= ζ a e γ ba ( ζ ∗ ,ζ )+ i R T dτ (cid:16) is ˙ ζζ ∗− ˙ ζ ∗ ζ | ζ | − H F ( ζ ∗ ,ζ ) (cid:17) = e − iωT ( s + s +1 / ) Z D µ ( ζ ) ζ ∗ ( T )= ζ ∗ b ζ (0)= ζ a exp ( γ ba ( ζ ∗ , ζ ) + i Z T dτ is ˙ ζζ ∗ − ˙ ζ ∗ ζ | ζ | − ω (2 s ) (cid:18) | ζ | | ζ | (cid:19) + ω s (2 s + 1) | ζ | | ζ | ! ) . (54)To proceed we shall use the Hubbard-Stratonovich [25–29] transformationwhich in our case can be realized through the introduction of the collectivefield ρ ( τ ) = | ζ | / (1 + | ζ | ). This can be consistently achieved [22] by makinguse of the functional identity1 = Z D ρ δ (cid:20) ρ − | ζ | | ζ | (cid:21) = Z D ρ Z D σe − i R T dτ (cid:18) ρ − | ζ | | ζ | (cid:19) σ . (55)Combining Eqs. (54) and (55) we can rewrite the correlator in the followingform: G F ( ζ ∗ b , ζ a ) = e − iωT ( s + s + ) Z D ρ Z D σ exp ( − iω Z T dτ ((2 s ) ρ − s ( s + 1) ρ ) − i Z T dτ ρσ ) ˜ G ( ζ ∗ b , ζ a ; σ ) , (56)where in this expression we wrote˜ G ( ζ ∗ b , ζ a ; σ ) = Z D µ ( ζ ) ζ ∗ ( T )= ζ ∗ b ζ (0)= ζ a e γ ba ( ζ ∗ ,ζ )+ i R T dτ (cid:18) is ˙ ζζ ∗− ˙ ζ ∗ ζ | ζ | + σ | ζ | | ζ | (cid:19) . (57)This integral has the same structure as the integral in Eq. (11) (with thechange ω → σ/ s ). Thus the calculation is straightforward and providedthat we adopt the symmetrized discrete definition of the functional integral11e get: ˜ G S ( ζ ∗ b , ζ a ; σ ) = (cid:16) ζ ∗ b ζ a e i s R T dτσ (cid:17) s e i s R T dτσ (1 + | ζ b | ) s (1 + | ζ a | ) s (58)= P sp =0 (cid:18) sp (cid:19) ( ζ ∗ b ζ a ) p e i s ( p + ) R T dτσ (1 + | ζ b | ) s (1 + | ζ a | ) s . Inserting this result into Eq. (56) we find: G F ( ζ ∗ b , ζ a ) = e − iωT ( s + s + )(1 + | ζ b | ) s (1 + | ζ a | ) s ×× s X p =0 (cid:18) sp (cid:19) ( ζ ∗ b ζ a ) p ×× Z D ρe − iω R T dτ ((2 s ) ρ − s ( s +1) ρ ) ×× Z D σe − i R T dτ ( ρ − s ( p + )) σ . (59)The last integral results to a delta function instructing that ρ = (1 / s )( p +1 / ρ is immediately performed yielding the exactresult: G F ( ζ ∗ b , ζ a ) = e − iωT ( s + s + )(1 + | ζ b | ) s (1 + | ζ a | ) s ×× s X p =0 (cid:18) sp (cid:19) ( ζ ∗ b ζ a ) p e − iωT ( p − sp ) = s X j = − s e − iωT j (2 s )!( s − j )!( s + j )! ×× ( ζ ∗ b ζ a ) s − j (1 + | ζ b | ) s (1 + | ζ a | ) s = s X j = − s e − iωT j h ζ b | s, j ih s, j | ζ a i . (60)For comparison we can repeat the previous steps beginning with the stan-dard Hamiltonian (53). In such a case the expression (56) assumes the form: G ( ζ ∗ b , ζ a ) = e − iωT s Z D ρ Z D σe − iω s ( s +1) R T dτ ( ρ − ρ ) × e − i R T dτρσ ˜ G ( ζ ∗ b , ζ a ; σ ) . (61)12ollowing the standard procedure for the evaluation of the last integral andtaking into account the “extra phase” contribution we find that:˜ G ( ζ ∗ b , ζ a ; σ ) = (cid:16) ζ ∗ b ζ a e i s R T dτσ (cid:17) s (1 + | ζ b | ) s (1 + | ζ a | ) s (62)= P sp =0 (cid:18) sp (cid:19) ( ζ ∗ b ζ a ) p e i s p R T dτσ (1 + | ζ b | ) s (1 + | ζ a | ) s . Once again the integral over σ forces ρ = p/ s and the amplitude (61) reads G ( ζ ∗ b , ζ a ) = e − iωT s (1 + | ζ b | ) s (1 + | ζ a | ) s s X p =0 (cid:18) sp (cid:19) ( ζ ∗ b ζ a ) p e − iωT s − s ( p − sp ) = s X j = − s e − iωT j + iωT j − s s h ζ b | s, j ih s, j | ζ a i . (63)As the comparison with the exact formula (47) proves, this is a wrong result.If we take, for example, the case s = 1 we get for the trace of the amplitude(63) the result already indicated in [21]Tr G = 2 e − iωT + e − iωT/ = Tr e − iT ˆ H . (64)At this point it would be enlightening to summarize our proposal forconstructing time continuous spin coherent state path integrals:Suppose that the dynamics of a spin system is described by a quantumHamiltonian of the form ˆ H = ˆ H ( ˆ S x , ˆ S y , ˆ S z ). The first step is to make useof the Holstein-Primakoff transformation (22) to rewrite the Hamiltonian interms of the bosonic creation and annihilation operators: ˆ H = ˆ H ( ˆ α † , ˆ α ).Next come the introduction of the quadratures ˆ q, ˆ p through the relationsˆ α = (ˆ q + i ˆ p ) √ , ˆ α † = (ˆ q − i ˆ p ) √
2. This step yields the recasting of theHamiltonian: ˆ H = ˆ H (ˆ p, ˆ q ). The crucial step is the third one consisting ofthe replacement of the quantum Hamiltonian by its classical counterpartˆ H → H F ( p, q ), where p and q are the representation of the quadratures inthe coherent state basis: p = h ζ | ˆ p | ζ i , q = h ζ | ˆ q | ζ i . As the analysis dictates,the resulting classical Hamiltonian must be used to define the continuumaction that weighs the summation of paths in the manifold we are workingwith.Note that the above recipe does not depend on the value of the spinof the system. This is the reason the above described road of Hamiltonianconstruction, is essentially the same for spin and bosonic systems as thelatter are the asymptotic limit of the former. Roughly speaking, the key ideais to begin from the well-defined phase-space Feynman path integral, andby making suitable canonical transformations to arrive at the description of13he problem we are interested in. This procedure also fixes the permissiblediscrete definition of the path integration.In the exactly solvable example discussed above, the final calculation hasbeen reduced to that of a simple harmonic oscillator. However, this is notneither the general case nor the most interested one. On the contrary, thecoherent-state path integration has been commonly used for semiclassicalcalculations. While the structure of this kind of calculations remain thesame in our approach, higher order differences arise due to the differentclassical action entering into the path integrals. As a concrete exampleconsider the Lipkin-Meshkov-Glik (LMG) model [35] in which the tunnelingsplitting has been quasi-classically calculated [14]:ˆ H = w √ s −
1) ( ˆ S + ˆ S − ) + sw √ , w > . (65)The “standard” form of this Hamiltonian reads as follows: h ζ | ˆ H | ζ i = √ sw ζ ∗ + ζ (1 + | ζ | ) + sw √ . (66)However, our recipe leads to the following form of the relevant classicalHamiltonian: H F ( ζ ∗ , ζ ) = √ s s − w ζ ∗ + ζ (1 + | ζ | ) (cid:18) | ζ | s (cid:19) + sw √ . (67)The two expressions coincide only at the asymptotic limit s → ∞ thus weexpect our approach to be more accurate when corrections of order 1 /s aresignificant. In a forthcoming work we shall present detail results on thesubject. In this work we present a method for defining and handling time-continuousspin coherent-sate path integral without facing inconsistencies. Such a path-integral formulation opens new possibilities for applying semiclassical ap-proximations with improved accuracy in quantum spin models. Moreover,this formalism can be also applied in bosonic systems, such as the two-siteBose-Hubbard model, ˆ H BH = − J ˆ S x + U ˆ S z at the large N limit [32–34],which are of increasing interest both theoretically and experimentally. Theaim of this paper is not the presentation of new results. It is, rather, anattempt to set a solid basis of reliable calculations.Our approach is based on two pylons. The first is the adoption ofa discretized form of the path integrals that is invariant under canonicaltransformations. The second is the three simple steps for identifying theHamiltonian that weighs the paths in the non-flat manifold ¯ C . In the first14tep we use the Holstein-Primakoff transformation to rewrite the quantumHamiltonian in terms of “position” and “momentum” operators. The sec-ond step consists of constructing the Feynman phase-space integral in whichthe classical version of the Hamiltonian enters. The third step is a canonicalchange of variables that produces the form of the Hamiltonian which entersinto the time-continuous spin coherent-state path integral. We have followedthis simple method to derive, directly in the continuum, the correct resultfor the simple case ˆ H ∼ ˆ S z for which inconsistencies have been repeatedlyreported. References [1] R.P. Feynman, Rev. Mod. Phys. , 367-387 (1948).[2] R.P. Feynman, Phys. Rev. , 108-128 (1951).[3] H. Kleinert: “Path Integrals in Quantum Mechanics, Statistics, Poly-mer Physics; and Financial Markets” World Scientific Publishing Co.,(2006).[4] R. J. Glauber, Phys.Rev. , 2766 (1963).[5] J. R. Klauder, “Path Integrals and their Applications in Quantum,Statistical and Solid State Physics”, edited by G. Papadopoulos and J.Devreese (Plenum Press, Antwerb, Belgium 1977)[6] F. A. Berezin, Sov. Phys. Usp. , 763 (1980).[7] J. R. Klauder and B. S. Skagerstam, “Coherent States, Applicationsin Physics and Mathematical Physics”, World Scientific, Singapore,(1985).[8] J. R. Klauder, Phys. Rev. D (8), 2349 (1979).[9] J. Kurchan, P. Leboeuf, and M. Saraceno, Phys. Rev. A , 6800 (1989).[10] H. G. Solari, J. Math. Phys. , 1097 (1987).[11] E. A. Kohetov, J. Math. Phys. , 4667 (1995).[12] E.A. Kohetov, J. Phys. A: Math. Gen. , 48(2003). 1515] H. Karatsuji, and Y. Mizobushi, J. Math. Phys. , 757 (1981).[16] A. Garg, and G.-H. Kim, Phys. Rev. B , 12921 (1992).[17] M. Enz, and R. Scilling, J. Phys. C L711 (1986).[18] M. Enz, and R. Scilling, J. Phys. C ,5633 (1997).[20] J. Shibata, and S. Takagi, Int. J. Mod. Phys. B , 107 (1999).[21] J.H. Wilson and V. Galitski , Phys. Rev. Lett. ,032104 (2014).[23] J. M. Radcliff, J. Phys. A: Gen. Phys. , 313 (1971).[24] J. Holstein, and H. Primakoff, Phys. Rev. , 1113 (1940).[25] R. Stratonovich, Sov. Phys. Dokl. , 416 (1957).[26] J. Hubbard, Phys. Rev. Lett. , 77 (1959).[27] B. Muhlshegel, J. Math. Phys. , 522 (1962).[28] M. B. Halpern, Nucl. Phys. B , 504 (1980).[29] A. Jevicki and B. Sakita, Nucl. Phys. B , 89 (1981).[30] H. Weyl, Z. Phys. , 1 (1927).[31] A. Polkovnikov, Ann. Phys. , 1790 (2010).[32] A.J. Leggett, Rev. Mod. Phys. , 307 (2001).[33] M. Holthaus, and S. Stenholm, Eur. Phys. J B , 451 (2001).[34] B. Juli´a-Di´az, T. Zibold, M. K. Oberthaler, M. Mel´e-Messeguer, J.Martorell, and A. Polls, Phys. Rev. A , 023615 (2012).[35] H. J. Lipkin, N. Meshkov and A. J. Glick, Nucl. Phys.62