Deformation quantization for systems with second-class constraints in deformed fermionic phase space

Deformation quan tization for systems with second-class constrain ts in deformed fermionic phase space Bing-Sheng Lin 1 , 2 , † , T ai-Hua Heng 3 1 Sc ho ol of Mathematics, South China Univ ersity of T echnology , Guangzhou 510641, China 2 Lab oratory of Quan tum Science and Engineering, South China Univ ersity of T echnology , Guangzhou 510641, China † Email: sclbs@scut.edu.cn 3 Sc ho ol of Ph ysics and Opto electronic Engineering, Anh ui Universit y , Hefei 230601, China 30 Ma y 2022 Abstract In order to quan tize systems inv olving second-class constraints, one should use Dirac brack et instead of Poisson brack et. F urthermore, one can sp ecify a star pro d- uct in which the term linear in ℏ is prop ortional to the Dirac brack et. In this wa y an oscillator system in a deformed fermionic phase space is analyzed and the cor- resp onding energy lev el and Wigner functions are ev aluated according to scheme of deformation quantization. W e also study the en tanglement entrop y induced by the deformation of the fermionic phase space. P ACS: 03.65.-w, 02.40.Gh, 05.30.Fk Keywor ds: deformed fermionic space; deformation quantization; en tanglement en tropy 1 In tro duction Usually there are three logically autonomous alternativ e approac hes to quantum me- c hanics. The first one is the standard op erator formalism in Hilb ert space, developed b y Heisenberg, Schr¨ odinger, Dirac and others in the t wen ties of the past century . The second one relies on path in tegral, and was conceived by Dirac and constructed by F eynman. The third one is deformation quantization formalism [1], based on Wigner’s quasi-distribution function [2] and W eyl’s corresp ondence b et ween quantum op erators and ordinary c -n umber phase space functions [3 – 8]. Ph ysical applications of the deformation quantization metho d hav e mainly b een re- stricted to systems inv olving b osonic degrees of freedom, both in quantum mechanics and quan tum field theory . Some researchers also studied fermionic algebras in quantum mec hanics by deformation quan tization metho ds [9 – 13]. These researches show that de- formation quantization is a p o w erful to ol for treating systems inv olving b osonic and/or fermionic degrees of freedom. There has b een muc h interest in the study of ph ysics in differen t kinds of deformed spaces, such as noncommutativ e space [14 – 20]. The ideas of noncommutativ e or de- formed space-time and field theories defined on suc h a structure started already in 1947 [21]. It came up again first in the 1980’s, when Connes form ulated the mathe- matically rigorous framew ork of noncommutativ e geometry [22]. In physical theories a noncomm utative space-time first app eared in string theory , namely in the quantization 1 of op en string [23 – 25]. Another field of in terest, where the noncommutativit y of space- time could pla y an important role, is quan tum gravit y [26, 27]. Also in condensed matter ph ysics the concept of noncommutativ e space-time is applied, which is relev ant for the in teger quan tum Hall effect [28]. On the other hand, b esides the bosonic co ordinate space whic h can b e deformed as mentioned ab o v e, the fermionic space can also b e deformed to the so-called non-anticomm utative space b y making the fermionic coordinates, i.e. the Grassmann v ariables, not an ticommuting, but satisfying a Clifford algebra. Seiberg sho wed this deformation ma y main tain the N = 1 / 2 sup ersymmetry and arise on branes in the presence of a graviphoton background [29]. Some other researchers also studied some types of deformed fermionic spaces and sup erspaces [30 – 35]. So it is significant to study ph ysical systems in a deformed fermionic space in deformation quantization. In this pap er we use deformation quan tization tec hnique to quantize pseudo classi- cal systems inv olving second-class constraints in deformed fermionic space. The pap er is organized as follo ws. Firstly , in section 2 we briefly summarize the pro cedure of quan tizing suc h a system in the deformed fermionic space according to the deformation quan tization. Then using this sc heme we quantize a system of t wo fermionic oscillators in a deformed fermionic space and obtain the energy lev el and corresp onding Wigner functions for the system in section 3. W e also study the entanglemen t entrop y of the ph ysical systems induced by the deformation of the fermionic spaces in section 4. As a comparison, in section 5 we also use op erator formalism in Hilb ert space to recalculate the entanglemen t en tropy of this system, and get the same result. The last section is conclusion. 2 Pseudo classical systems in deformed fermionic space and their quan tization W e consider a pseudo classical system with fermionic degrees of freedom, whic h describ ed b y Grassmann v ariables θ α . The Grassmann parity of a quantit y f is denoted b y ε ( f ), for example, ε ( θ α ) = 1 and ε ( θ α θ β ) = 0. The Lagrangian is a function of the Grassmann co ordinates and their velocities L ( θ α , ˙ θ β ), α, β = 1 , ..., n . The canonical conjugate mo- men ta are π α = − → ∂ L/∂ ˙ θ α , where − → ∂ /∂ ˙ θ denotes left deriv atives for Grassmann v ariables. Th us we ha v e a 2 n -dimensional o dd (or Grassmann) phase space M , and a p oint z in M is wri tten as z = ( θ α , π β ). The Hamilton function is (here sum conv ention is understoo d) H ( θ α , π β ) = ˙ θ α π α − L. (1) The Poisson brac ket { F , G } of tw o functions F and G on M is a Z 2 -graded bilinear map with the follo wing prop erties: (i) { F , G } = − ( − ) ε ( F ) ε ( G ) { G, F } ; (ii) { F , GH } = { F , G } H + ( − ) ε ( F ) ε ( G ) G { F , H } ; (iii) {{ F , G } , H } + ( − ) ε ( F )( ε ( G )+ ε ( H )) {{ G, H } , F } + ( − ) ε ( H )( ε ( F )+ ε ( G )) {{ H , F } , G } = 0. In the ordinary o dd phase space M the P oisson brac ket may be written in terms of canonical co ordinates as { F , G } = − ( − ) ε ( F ) − → ∂ F ∂ θ α − → ∂ G ∂ π α + − → ∂ F ∂ π α − → ∂ G ∂ θ α ! , (2) and w e hav e the canonical relations { θ α , π β } = δ αβ , (3) 2 the brac kets inv olving tw o co ordinates or t wo momentum v ariables v anish. The Hamil- ton equations gov erning ev olution of the corresp onding pseudo classical system ma y b e written as d F d t = { F , H } . (4) W e may abbreviate our notation in Eq. (2) in terms of the co ordinate z = ( z 1 , ..., z 2 n ) in the 2 n -dimensional o dd phase space M and a Poisson tensor A ij , where the indices i, j run from 1 to 2 n . In canonical v ariables, from Eq. (2) the A can b e read as A =  0 I n I n 0  , where I n is the n × n unit matrix. Then Eq. (2) b ecomes { F , G } = − A ij F ← − ∂ z i − → ∂ z j G, (5) where the relation b et w een the left and right deriv atives of functions of Grassmann v ariables has b een used − → ∂ ∂ θ α F = ( − ) ε ( F ) F ← − ∂ ∂ θ α . (6) In these notations w e hav e − → ∂ ∂ θ α θ β = δ αβ = − θ β ← − ∂ ∂ θ α . Noticing that the ab o ve simple expression of A only w orks for canonical v ariables and in an ordinary an ticommutativ e o dd phase space. In more complicated case, from the prop ert y (i) of the P oisson brac kets we realize that the Poisson tensor matrix A for the o dd phase space m ust b e symmetric. Therefore we ma y use a suitable symmetric matrix A to reflect an ticommutativit y of a deformed fermionic phase space, and define the corresp onding P oisson brack et of tw o functions F and G on suc h a deformed fermionic phase space as { F , G } df = −A ij F ← − ∂ z i − → ∂ z j G. (7) Of course, suc h defined Poisson brack ets must satisfy the ab o ve prop erties (i), (ii) and (iii). According to this definition, the canonical relations b et ween θ α , π β should b e differen t from Eq. (3) in general to reflect the comm utativity of the deformed fermionic phase space, and the Hamilton equation of motion for a pseudo classical system on such a deformed fermionic phase space is still d F d t = { F , H } df . (8) F or a classical system in volving constraints additional concepts arise. Usually the constrain ts can b e divided into tw o classes. The first-class constraints are those whose P oisson brac k ets with all other constraints v anish, and the remaining constrain ts are second-class. According to Dirac [36], to describ e systems with second-class constraints χ α , α = 1 , ..., l in the deformed fermionic phase space, one should use Dirac brac kets instead of the P oisson brack ets, { F , G } D = { F , G } df − { F, χ α } df C − 1 αβ { χ β , G } df , (9) where C αβ = { χ α , χ β } df and C − 1 αβ is the matrix inv erse to C αβ . Then the second-class constrain ts should b e set strongly to zero. In order to quantize the pseudo classical systems living in suc h a deformed fermionic phase space, by the pro cedure of deformation quantization, the ordinary m ultiplication of the pseudo classical functions on the deformed fermionic phase space should b e replaced b y a deformed asso ciative ∗ -pro duct. The ∗ -product should be c hosen in such a w ay so that the term linear in the deformation parameter (for example, ℏ ) in the ∗ -pro duct is proportional to the Poisson brack et. In the case of systems inv olving second-class 3 constrain ts the term linear in ℏ in the ∗ -pro duct should b e prop ortional to the Dirac brac ket instead of the Poisson brack et. The ∗ -pro duct on a given phase space is in the b est case unique only up to an equiv alence class. Equiv alen t ∗ -pro ducts corresp ond to differen t quantization schemes. Usually one use the Moy al ∗ -pro duct, whose general form in the deformed fermionic phase space (when the second-class constrain ts do not in volv e) should b e F ∗ G = F exp  − i ℏ 2 A ij ← − ∂ z i − → ∂ z j  G. (10) The so-called time-ev olution function Exp( − i H t/ ℏ ) is the solution of the equation i ℏ d d t Exp( − i H t/ ℏ ) = H ∗ Exp( − i H t/ ℏ ) , (11) whic h is the equiv alent of the time-dep enden t Sch¨ odinger equation in this context. F or a time-indep enden t Hamilton function the solution is given by the star exp onen tial Exp( − i H t/ ℏ ) = ∞ X n =0 1 n !  − i t ℏ  n H n ∗ , (12) where H n ∗ is the n -th ∗− p o w er of the function H , H n ∗ = H ∗ H ∗ ... ∗ H | {z } n . (13) The time-ev olution function has a F ourier-Dirichlet expansion of the form [8] Exp( − i H t/ ℏ ) = X E W E e − i E t/ ℏ , (14) the W E are Wigner functions corresp onding to the energy E , whic h satisfy the ∗ -genv alue equation H ∗ W E = E W E = W E ∗ H . (15) This ∗ -gen v alue equation is the equiv alent of the time-indep enden t Sc hr¨ odinger equation in this con text. The sp ectral decomp osition of the Hamilton function is given by H = X E E W E . (16) The Wigner functions are idemp oten t and complete W E ∗ W E ′ = δ E ,E ′ W E , X E W E = 1 . (17) In the following sections w e show the concrete pro cedures of deformation quantization b y discussing a simple example of tw o fermionic oscillators in a sp ecial type of deformed fermionic space. 3 Tw o fermionic oscillators in a deformed fermionic space Some authors are in terested in a case that the anticomm uting v ariables form a Clifford algebra lik e [29 – 35] { θ α , θ β } ∗ = c αβ , (18) where { F , G } ∗ ≡ F ∗ G + G ∗ F is a star anticomm utator, and this deformed algebra ma y arise in string theory in a graviphoton bac kground. W e can call this space non- an ticommutativ e (NA C) space. 4 Consider a fermionic system in the NAC space inv olves four Grassmann co ordinates θ α , α = 1 , 2 , 3 , 4. The Lagrange function is L = i 2 ( θ 1 ˙ θ 1 + θ 2 ˙ θ 2 + θ 3 ˙ θ 3 + θ 4 ˙ θ 4 ) + i ω θ 1 θ 3 + i ω θ 2 θ 4 , (19) whic h describ es a system of tw o fermionic oscillators in the NA C space. The canonical momen ta are π α = − i 2 θ α , so that the constraints are χ α = π α + i 2 θ α , and the Hamilton function is giv en by H = ˙ θ α π α − L = − i ω θ 1 θ 3 − i ω θ 2 θ 4 . (20) F or example, one can use the following P oisson brack et in the non-anticomm utative space, { F , G } nac = − F h ← − ∂ θ α − → ∂ π α + ← − ∂ π α − → ∂ θ α +i C  ← − ∂ θ 1 − → ∂ θ 2 + ← − ∂ θ 2 − → ∂ θ 1 + ← − ∂ θ 3 − → ∂ θ 4 + ← − ∂ θ 4 − → ∂ θ 3  i G, (21) where C is some parameter. It is easy to v erify that, this Poisson brack et satisfies the prop erties (i), (ii) and (iii) in the previous section. Th us the Poisson brack ets of the constrain ts are { χ α , χ β } nac = i δ αβ − i C 4 ( δ α 1 δ β 2 + δ α 2 δ β 1 + δ α 3 δ β 4 + δ α 4 δ β 3 ) , (22) whic h mean that the constraints are second-class, and they can b e expressed by a matrix C αβ = { χ α , χ β } nac = i     1 − C 4 0 0 − C 4 1 0 0 0 0 1 − C 4 0 0 − C 4 1     (23) with its in verse C − 1 αβ = − i 1 − C 2 16     1 C 4 0 0 C 4 1 0 0 0 0 1 C 4 0 0 C 4 1     . (24) Using Eq. (9) w e obtain the Dirac brack ets in this context, { F , G } D = 4i 1 − C 2 16 F  ← − ∂ θ 1 − → ∂ θ 1 + ← − ∂ θ 2 − → ∂ θ 2 + ← − ∂ θ 3 − → ∂ θ 3 + ← − ∂ θ 4 − → ∂ θ 4  G + i C 1 − C 2 16 F  ← − ∂ θ 1 − → ∂ θ 2 + ← − ∂ θ 2 − → ∂ θ 1 + ← − ∂ θ 3 − → ∂ θ 4 + ← − ∂ θ 4 − → ∂ θ 3  G. (25) Here we hav e implemented the constrain ts as strong equations according to Dirac’s metho d [36] so the only indep enden t v ariables are then the θ α . F rom now on we use the Dirac brac kets instead of the Poisson brack ets in this NAC space. As mentioned in section 2, in order to quantize a classical system the Mo yal star pro duct should b e c hosen so that its first-order term is proportional to the Dirac brac ket in the case inv olving second class constraints. Th us w e use the following Moy al star pro duct for the NAC space ∗ ′ = exp  1 2 ← − ∂ θ α A αβ − → ∂ θ β  , α, β = 1 , 2 , 3 , 4 , (26) 5 where the symmetric matrix A is A =     ℏ c 0 0 c ℏ 0 0 0 0 ℏ c 0 0 c ℏ     , (27) with c = ℏ C / 4. It is easy to verify that the indep enden t v ariables satisfy the following star an ticommutators, { θ i , θ i } ∗ ′ = ℏ , { θ 1 , θ 2 } ∗ ′ = { θ 3 , θ 4 } ∗ ′ = c, i = 1 , 2 , 3 , 4 , (28) and others v anish. F urthermore, in the present work, we will consider a more general case, in which the indep enden t v ariables satisfying the following star anticomm utators { θ i , θ i } ∗ = ℏ , { θ 1 , θ 2 } ∗ = c , { θ 3 , θ 4 } ∗ = d, i = 1 , 2 , 3 , 4 , (29) and others v anish. The parameters c, d are real n umbers, and usually we also assume | c | , | d | ≪ ℏ . The corresp onding star pro duct is F ∗ G = F exp  ℏ 2  ← − ∂ θ 1 − → ∂ θ 1 + ← − ∂ θ 2 − → ∂ θ 2 + ← − ∂ θ 3 − → ∂ θ 3 + ← − ∂ θ 4 − → ∂ θ 4  + c 2  ← − ∂ θ 1 − → ∂ θ 2 + ← − ∂ θ 2 − → ∂ θ 1  + d 2  ← − ∂ θ 3 − → ∂ θ 4 + ← − ∂ θ 4 − → ∂ θ 3   G. (30) This means that the fermionic star pro duct leads to a Cliffordization of the Grassmann algebra of the fermionic v ariables, and it is a sp ecial realization of Eq. (18). In this stage one finds out that if the deformation parameters c = d = 0, the Hamilton function (20) describ es tw o indep enden t fermionic oscillators. F or conv enience, one can divide the Hamilton function (20) into tw o parts as follow, H = H + + H − , (31) where H + = − i ω 2 ( θ 1 θ 3 + θ 2 θ 4 + θ 1 θ 4 + θ 2 θ 3 ) H − = − i ω 2 ( θ 1 θ 3 + θ 2 θ 4 − θ 1 θ 4 − θ 2 θ 3 ) . (32) It is easy to v erify that H + ∗ H − = H + H − = H − H + = H − ∗ H + , (33) and H ± ∗ H ± = h 2 ± ω 2 4 . (34) where h ± = p ( ℏ ± c )( ℏ ± d ). After some straigh tforward calculations, one can obtain i h ± d d t Exp( − i H ± t/h ± ) = H ± ∗ Exp( − i H ± t/h ± ) , (35) where Exp( − i H ± t/h ± ) are the time-ev olution functions, Exp( − i H ± t/h ± ) = ∞ X n =0 1 n !  − i t h ±  n ( H ± ) n ∗ = W + ± e − i ωt 2 + W − ± e i ωt 2 . (36) 6 and W + ± = 1 2 + H ± h ± ω , W − ± = 1 2 − H ± h ± ω . (37) The corresp onding ∗ -eigenv alue equations are H ± ∗ W i ± = E i ± W i ± = W i ± ∗ H ± , i = + , − , (38) and E + ± = h ± ω 2 , E − ± = − h ± ω 2 , (39) The sp ectral decomp osition of the Hamilton function is given by H ± = E + ± W + ± + E − ± W − ± . (40) The Wigner functions are idemp oten t and complete, W i ± ∗ W j ± = δ ij W i ± , W + ± + W − ± = 1 . (41) W e also hav e W i + ∗ W j − = W i + W j − = W j − W i + = W j − ∗ W i + . (42) F or the total Hamiltonian (20), the corresp onding Wigner functions are just W ij ≡ W i + ∗ W j − , i, j = + , − , (43) and these Wigner functions satisfy W ij ∗ W kl = δ ik δ j l W ij , (44) The ∗ -eigen v alue equation is H ∗ W ij = E ij W ij = W ij ∗ H , (45) where E ij = E i + + E j − are the energy lev el of the total system. F or example, W ++ = W + + ∗ W + − = 1 4 + H + 2 h + ω + H − 2 h − ω + H + H − h + h − ω 2 = 1 4 − i h + + h − 4 h + h − ( θ 1 θ 3 + θ 2 θ 4 ) + i h + − h − 4 h + h − ( θ 1 θ 4 + θ 2 θ 3 ) + θ 1 θ 2 θ 3 θ 4 h + h − . (46) W e also hav e [37] T r( W ij ) = 4 ℏ 4 Z dθ 4 dθ 3 dθ 2 dθ 1 ( ⋆W ij ) = 1 , (47) where ⋆F is the Ho dge dual for Grassmann monimials, which maps a Grassmann mono- mial with grade r into a Grassmann monomial with grade d − r , and d is the num b er of Grassmann basis elemen ts, ⋆ ( θ i 1 θ i 2 ...θ i r ) = 1 ( d − r )! ε i r +1 ...i d i 1 i 2 ...i r θ i r +1 ...θ i d , 1 ⩽ r ⩽ d. (48) The in tegration is given by the Berezin integral for whic h we hav e R dθ i θ j = ℏ δ ij . 7 4 En tanglemen t of the fermionic oscillators in the NA C space. As w e hav e already known [38 – 40], in noncommutativ e b osonic spaces, the noncommu- tativit y of the spaces can induce some types of entanglemen t of the physical systems. So it is significant to study the en tanglement of physical systems in deformed fermionic spaces. In phase spaces, the quan tum R´ enyi entrop y can b e defined as [41], S α ( W ) = 1 1 − α ln (T r( W α ∗ )) , (49) where α  = 1 is some p ositiv e parameter. In the limit for α → 1, the quantum R´ en yi en tropy is just the von Neumann entrop y , S 1 ( W ) = − T r( W ∗ ln ∗ W ) , (50) where the ∗− logarithm is ln ∗ ( f ) := − ∞ X n =1 (1 − f ) n ∗ n . (51) F rom (44) and (47), one can obtain S α ( W ij ) = 0 , (52) whic h means that the states W ij are all pure states. By virtue of the partial entanglemen t en tropy , the entanglemen t measure of W ij can b e defined as the quantum entrop y of the reduced states W (1) ij ( θ 1 , θ 3 ) or W (2) ij ( θ 2 , θ 4 ). As an example, let us consider the Wigner function W ++ (46). The corresp onding Wigner functions of reduced states in the subspaces ( θ 1 , θ 3 ) and ( θ 2 , θ 4 ) are W (1) ++ ( θ 1 , θ 3 ) = T r θ 2 θ 4 ( W ++ ) = 2 ℏ 2 Z dθ 4 dθ 2 ( ⋆W ++ ) = 1 2 − i h + + h − 2 h + h − θ 1 θ 3 , (53) where T r θ 2 θ 4 is the partial trace o ver θ 2 , θ 4 , and W (2) ++ ( θ 2 , θ 4 ) = T r θ 1 θ 3 ( W ++ ) = 2 ℏ 2 Z dθ 3 dθ 1 ( ⋆W ++ ) = 1 2 − i h + + h − 2 h + h − θ 2 θ 4 . (54) Since the Wigner functions are just quasi-probabilit y distributions and they can hav e negativ e eigenv alues. One can v erify that, using the normal definition (49), the v alues of quan tum R´ enyi entrop y of the ab ov e reduced state Wigner functions (53) and (54) can b e negative. In some cases one can ev en obtain complex num b er results. So one should not use the normal definition of von Neumann entrop y in this case. Similar to Ref. [42], instead one can use the abstract v alues of the eigenv alues p i of the Wigner functions to define the quan tum entrop y , S ( W ) = − X i | p i | ln | p i | . (55) No w let us define the following functions f 1 = 1 2 − i ℏ θ 1 θ 3 , f 2 = 1 2 + i ℏ θ 1 θ 3 . (56) 8 It is easy to v erify that these functions are complete and orthonormal, X i =1 , 2 f i = 1 , f i ∗ f j = δ ij f i . (57) After some straigh tforward calculations, one can obtain W (1) ++ ( θ 1 , θ 3 ) ∗ f i = p i f i , (58) where p 1 = 1 2 + ℏ h + + h − 4 h + h − , p 2 = 1 2 − ℏ h + + h − 4 h + h − . (59) So f i are the eigenfunctions of W (1) ++ ( θ 1 , θ 3 ), and p i are the corresp onding eigenv alues. It is easy to see that the ab o ve eigenv alues p 1 + p 2 = 1, and p 1 ⩾ 1, p 2 ⩽ 0. This is b ecause the Wigner functions are just quasi-probability distributions and they can hav e negativ e eigenv alues. So the partial en tanglement entrop y E p of the state W ++ is E p ( W ++ ) ≡ S  W (1) ++ ( θ 1 , θ 3 )  = −| p 1 | ln | p 1 | − | p 2 | ln | p 2 | = −     1 2 + ℏ h + + h − 4 h + h −     ln     1 2 + ℏ h + + h − 4 h + h −     −     1 2 − ℏ h + + h − 4 h + h −     ln     1 2 − ℏ h + + h − 4 h + h −     = −  1 2 + ℏ h + + h − 4 h + h −  ln  1 2 + ℏ h + + h − 4 h + h −  −  ℏ h + + h − 4 h + h − − 1 2  ln  ℏ h + + h − 4 h + h − − 1 2  . (60) One can also calculate the entanglemen t entrop y of the states W −− , W + − and W − + . There are E p ( W −− ) = E p ( W ++ ) = −  1 2 + ℏ h + + h − 4 h + h −  ln  1 2 + ℏ h + + h − 4 h + h −  −  ℏ h + + h − 4 h + h − − 1 2  ln  ℏ h + + h − 4 h + h − − 1 2  , E p ( W + − ) = E p ( W − + ) (61) = −  1 2 + ℏ h + − h − 4 h + h −  ln  1 2 + ℏ h + − h − 4 h + h −  −  1 2 − ℏ h + − h − 4 h + h −  ln  1 2 − ℏ h + − h − 4 h + h −  . Here w e alwa ys assume that | c | , | d | ≪ ℏ . When c = d , w e hav e h ± = ℏ ± c . There are E p ( W ++ ) = E p ( W −− ) = − 2 ℏ 2 − c 2 2( ℏ 2 − c 2 ) ln 2 ℏ 2 − c 2 2( ℏ 2 − c 2 ) − c 2 2( ℏ 2 − c 2 ) ln c 2 2( ℏ 2 − c 2 ) , E p ( W + − ) = E p ( W − + ) = − ℏ 2 + ℏ c − c 2 2( ℏ 2 − c 2 ) ln ℏ 2 + ℏ c − c 2 2( ℏ 2 − c 2 ) − ℏ 2 − ℏ c − c 2 2( ℏ 2 − c 2 ) ln ℏ 2 − ℏ c − c 2 2( ℏ 2 − c 2 ) . (62) When c = − d , w e hav e h + = h − = √ ℏ 2 − c 2 , and E p ( W ++ ) = E p ( W −− ) = − ℏ + √ ℏ 2 − c 2 2 √ ℏ 2 − c 2 ln ℏ + √ ℏ 2 − c 2 2 √ ℏ 2 − c 2 − ℏ − √ ℏ 2 − c 2 2 √ ℏ 2 − c 2 ln ℏ − √ ℏ 2 − c 2 2 √ ℏ 2 − c 2 , E p ( W + − ) = E p ( W − + ) = ln 2 . (63) 9 The second result ab o ve means that the states W + − and W − + are alw ays maximally en tangled states with any v alues c = − d . F or the special case c = d = 0, there are h + = h − = ℏ . So E p ( W ++ ) = E p ( W −− ) = 0 and E p ( W + − ) = E p ( W − + ) = ln 2 = 0 . 693. This means that the states W ++ and W −− ha ve no entanglemen t. This returns to the case of tw o indep enden t fermionic oscillators in normal comm utative fermionic space. As an example, the entanglemen t entrop y E p of the states W ij in the case c = d are plotted in Figs. 1 and 2. - 0.3 - 0.2 - 0.1 0.1 0.2 0.3 c ℏ 0.02 0.04 0.06 0.08 0.10 E p Figure 1: In the case c = d , the en tanglement en tropy E p ( W ++ ) = E p ( W −− ) with resp ect to the v ariable c/ ℏ . - 0.3 - 0.2 - 0.1 0.1 0.2 0.3 c ℏ 0.65 0.66 0.67 0.68 0.69 E p Figure 2: In the case c = d , the en tanglement en tropy E p ( W + − ) = E p ( W − + ) with resp ect to the v ariable c/ ℏ . Ob viously , in the case c = d , the entanglemen t entrop y E p ( W ++ ) = E p ( W −− ) b e- comes larger as the absolute v alue of c increases. This means that the entanglemen t of the states W ++ and W −− increases with the increase of the noncomm utativity of the fermionic phase space. On the contrary , the entanglemen t entrop y E p ( W + − ) = E p ( W − + ) b ecomes smaller as the absolute v alue of c increases. This means that the entanglemen t of the states W + − and W − + decreases with the increase of the noncomm utativity of the fermionic phase space. It is worth mentioning that, in noncommutativ e b osonic/fermionic phase spaces, the noncommutativ e co ordinates can b e transformed into those in normal commutativ e 10 phase space via the so-called Bopp’s shift (or Seib erg-Witten maps) [43], and one can solv e the new ph ysical systems with transformed coordinates in normal comm utative spaces instead of the original ones in noncomm utative spaces. Similarly , one can also use such kinds of transformations in the deformed fermionic space ( θ 1 , θ 2 , θ 3 , θ 4 ), and then study the entrop y and entanglemen t of the tw o fermionic oscillator system. But usually such kinds of transformations are not unitary , which may not preserve entropies of the subsystems. So it ma y change the entanglemen t of the original system. 5 Op erator formalism in Hilb ert space As a comparison, one can also use op erator formalism in Hilb ert space to analyse this system. It is w ell known that b osonic Wigner op erator maybe written as [2] ˆ ∆ b ( x, p ) = 1 (2 π ℏ ) 2 Z d u d v exp  i u ℏ ( ˆ p − p ) + i v ℏ ( ˆ x − x )  , (64) where ˆ x , ˆ p are ordinary co ordinate and momen tum op erators (which satisfies commuta- tion relation [ ˆ x, ˆ p ] = i ℏ ), and x and p are classical canonical co ordinate and momentum v ariables resp ectiv ely . In Eq. (64) u and v are real integral v ariables from −∞ to ∞ . Ha ving the ab o ve b osonic Wigner op erator, for a classical function h ( x, p ) of the canon- ical v ariables x and p , one may get its quantum corresp ondence ˆ H ( ˆ x, ˆ p ) = Z d p d x h ( x, p ) ˆ ∆ b ( x, p ) . (65) F or example, if the classical function is xp , according to Eq. (65) one gets its quan tum corresp ondence ( ˆ x ˆ p + ˆ p ˆ x ) / 2, whic h is the W eyl corresp ondence. Using b osonic annihila- tion and creation op erators ˆ b = ( ˆ x + i ˆ p ) / √ 2 ℏ and ˆ b † = ( ˆ x − i ˆ p ) / √ 2 ℏ , and holomorphic v ariables z = ( x + i p ) / √ 2 ℏ , for complex in tegral v ariable w = ( u − i v ) / √ 2, the ab o ve b osonic Wigner op erator can b e written as ˆ ∆ b ( z ∗ , z ) = 1 (2 π ℏ ) 2 Z d w ∗ d w exp  w ∗ √ ℏ ( ˆ b − z ) − ( ˆ b † − z ∗ ) w √ ℏ  . (66) F or a single-mode fermion, in the operator formalism the corresp onding Hilb ert space is tw o-dimensional. The creation and annihilation op erators ˆ a † and ˆ a of a single-mo de fermion satisfy the follo wing anticomm utation relations { ˆ a, ˆ a † } ≡ ˆ a ˆ a † + ˆ a † ˆ a = 1 , ˆ a 2 = ˆ a † 2 = 0 . (67) Let | 0 ⟩ b e the v acuum state ( ˆ a | 0 ⟩ = 0), then there is only one excitation state | 1 ⟩ = ˆ a † | 0 ⟩ . The Grassmann parities are ε ( | 0 ⟩ ) = 0 and ε ( | 1 ⟩ ) = 1. The op erators ˆ a and ˆ a † can also b e expressed as ˆ a = | 0 ⟩⟨ 1 | , ˆ a † = | 1 ⟩⟨ 0 | . The n umber op erators ˆ a † ˆ a acting on these states lead to ˆ a † ˆ a | 0 ⟩ = 0, ˆ a † ˆ a | 1 ⟩ = | 1 ⟩ . The pseudo classical corresp ondences of the op erators ˆ a and ˆ a † are denoted as η and η ∗ resp ectiv ely , which are the Grassmann num b ers. Similar to the b osonic case, the Wigner op erator of a single-mo de fermion can b e written as [44, 45] ˆ ∆ f ( η ∗ , η ) = Z d ξ ∗ d ξ exp  ξ ∗ (ˆ a − η ) − (ˆ a † − η ∗ ) ξ  = 1 2 − ( ˆ a † − η ∗ )(ˆ a − η ) , (68) where ξ is a complex Grassmann integral v ariable. The quantum corresp ondence of a pseudo classical function h ( η ∗ , η ) of the holomorphic Grassmann v ariables η ∗ and η is ˆ H (ˆ a † , ˆ a ) = Z d η ∗ d η h ( η ∗ , η ) ˆ ∆ f ( η ∗ , η ) . (69) 11 F or example, by virtue of integration laws of the Grassmann v ariables, one finds that the op erators ˆ a and ˆ a † are the quantum corresp ondences of the holomorphic v ariables η and η ∗ , resp ectiv ely . No w let us return to discuss the system analyzed in the previous section. One ma y in tro duce the following holomorphic v ariables for our fermionic system in the NA C space η = 1 √ 2 ℏ ( θ 1 + i θ 3 ) , η ∗ = 1 √ 2 ℏ ( θ 1 − i θ 3 ) , (70) whic h both are Grassmann v ariables also. F or the reduced state Wigner function W (1) ++ ( θ 1 , θ 3 ), there is W (1) ++ ( θ 1 , θ 3 ) = 1 2 − i h + + h − 2 h + h − θ 1 θ 3 = 1 2 − ℏ h + + h − 2 h + h − η ∗ η = W (1) ++ ( η ∗ , η ) . (71) Using Eqs. (68) and (69), one can get its op erator formalism, ˆ W (1) ++ = Z d η ∗ d η W (1) ++ ( η ∗ , η ) ˆ ∆ f ( η ∗ , η ) = 1 2 − ℏ h + + h − 2 h + h −  ˆ a † ˆ a − 1 2  . (72) Th us the tw o states | 0 ⟩ , | 1 ⟩ are eigenstates of the ab o v e op erator (72), and the corre- sp onding eigenv alues are λ 1 = 1 2 + ℏ h + + h − 4 h + h − , λ 2 = 1 2 − ℏ h + + h − 4 h + h − , (73) whic h exactly corresp onds to the result in Eq. (59). One can also use the matrix representations, | 0 ⟩ =  1 0  , | 1 ⟩ =  0 1  , (74) and ˆ a = | 0 ⟩⟨ 1 | =  0 1 0 0  , ˆ a † = | 1 ⟩⟨ 0 | =  0 0 1 0  . (75) So the op erator (72) is just the following diagonal matrix, ˆ W (1) ++ = 1 2 + ℏ h + + h − 4 h + h − 0 0 1 2 − ℏ h + + h − 4 h + h − ! =  λ 1 0 0 λ 2  . (76) It is easy to see that, for c, d  = 0, there should b e λ 2 < 0 < 1 < λ 1 . So one can not use the normal formula of the von Neumann entrop y . Instead, the quan tum en tropy of the system can b e defined as S ( W ) = − P i | λ i | ln | λ i | . 6 Conclusion In this pap er w e illustrate how to quan tize systems living in a deformed fermionic space according to the procedure of deformation quantization. Sometimes this kind of systems in volv es second-class constraints and in this case the Dirac brack et is useful to replace the Poisson brack et so that the corresp onding ∗ -pro duct can b e sp ecified. W e discuss a system of t wo fermionic oscillators in a so-called non-anticomm utative space, which in volv es the second-class constraints. By virtue of deformation quan tization metho ds, 12 w e obtain the Wigner functions and the corresp onding energy lev el of this system. W e also study the en tanglement induced by the deformation of the fermionic phase space. As a comparison, w e also use more p opular op erator formalism in Hilb ert space to obtain the same results, whic h convince ourself that the ab o ve results are correct. These efforts sho w that the deformation quantization is a p o werful to ol to deal with not only systems living in ordinary commutativ e space, but also those living in deformed b osonic or fermionic space. 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