š þfnø ez K Döf HmŒÆÔnÆ 2012c11 20F

Transcription

1 š þfnø ez K Döf HmŒÆÔnÆ 01c110F

2 K8¹Â 1 Dڹ 3DÚþfnØ š M îéaññxú"dum î¹kjü UþŠÏ~ Eê AÇØÅð" #¹Â š M  ãguþfnx E,3DÚþf nø Ä5Ÿ =µ 1 Š ê ke =3 ½Ä SÈ ½ 3AÇ)º 3 ±žmüzn5 =AÇÅð"

3 1 â» 5 #nøg Ônþ Š ê ŽÎ 5 7 ^ " Œ±n) ŽÎ 5 êæ^ Lr" Ì gž ï 5 ƒ±äkôn ½f^ ~ XµPTé 5½ é 5 ѽ " 3 ïäyg Eˆ«š äkpté 5½ é 5 M îxú ) êš!½â#½sè! Ûž müzn5" 4 óš E#. äkpt é 5 ѽ ÛT. é 59Ùgu»"3 ) êš åš^"

4 Spontaneous breaking of permutation symmetry in pseudo-hermitian quantum mechanics collaborated with Jun-Qing Li School of Physics, Nankai University (Phys. Rev. A 85 (01) [arxiv: [quant-ph]]) November 0, 01

5 Main content 1 Backgrounds Our model and some properties of pseudo-hermitian systems 3 Equivalence of our model and Pais-Uhlenbeck oscillator 4 Permutation and its spontaneous breaking in our model 5 The identity and its spontaneous breaking in the Pais-Uhlenbeck oscillator 6 Conclusion

6 Main content 1 Backgrounds Our model and some properties of pseudo-hermitian systems 3 Equivalence of our model and Pais-Uhlenbeck oscillator 4 Permutation and its spontaneous breaking in our model 5 The identity and its spontaneous breaking in the Pais-Uhlenbeck oscillator 6 Conclusion

7 Backgrounds In 1940 s, Dirac, Pauli, proposed a non-hermitian Hamiltonian and indefinite metric in Hilbert space, in order to solve divergence and relative problems. (Dirac, Proc. Roy. Soc. Lond. A 180 (194) 1; Pauli, Rev. Mod. Phys. 15 (1943) 175) Ú\ 5 Žfη ؽÝ5Žf òï~sè φψdq í φηψdq Ïd M îhäkƒagš 5µH = H := η 1 H η yônœ*ÿþ êµ H Av = H Av Ù H Av := ψηhψdq.

8 Backgrounds In 1969, Lee and Wick, used the above idea in QED to keep the unitarity of S-matrix. (Lee and Wick, Nucl. Phys. B 9 (1969) 09) From 1970 s to 1998, some literature revealed that a non-hermitian Hamiltonian could have real eigenvalues under specific conditions, but it is not a hot topic. (R. Brower, M. Furman and M. Moshe, Phys. Lett. B 76 (1978) 13; B.C. Harms, S.T. Jones and C.-I Tan, Nucl. Phys. B 171 (1980) 39; Phys.Lett. B 91 (1980) 91; E. Caliceti, S. Graffi and M. Maioli, Comm. Math. Phys. 75 (1980) 51; A.A. Andrianov, Ann. Phys. (N.Y.) 140 (198) 8; T. Hollowood, Nucl. Phys. B 384 (199) 53 [arxiv:hep-th/ ]; F.G. Scholtz, H.B. Geyer and F.J.W. Hahne, Ann. Phys. (N.Y.) 13 (199) 74, etc.)

9 Backgrounds In 1998, C.M. Bender and S. Boettcher, (C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80 (1998) 543), proposed a class of PT -symmetric Hamiltonians that can have real spectrum. (P: parity, T: time reversal) H = p + m x (ix) N, Ù N ê Ø uê"n = ž T. òz ÊÏf"w, ù.3xept C e ±ØCµ P : x x, p p, i +i; T : x +x, p p, i i. êšož(jl²µ1 N ž UÌ Ú 1 < N < ž k êþ ÚUþ Š à õuþš EÝé 3 N 1ž vk êuþš"

10 Backgrounds In 00 and 003, A. Mostafazadeh, (A. Mostafazadeh, J. Math. Phys. 43, 05 (00); 43, 3944 (00); 44, 974 (003)), developed the pseudo-hermitian theory, e.g. introduced the associated positive definite inner product. From 1998 to now The idea of non-hermitian or pseudo-hermitian Hamiltonians has appeared in various regions of physics. For instance, QFT (C.M. Bender, et al., arxiv:hep-th/040011) supersymmetric QM and QFT (C.M. Bender and K.A. Milton, arxiv:hep-th/ ) NCQFT (Y.-G. Miao, et al., arxiv:hep-th/ ) biological physics (B. Eslami-Mossallam, M.R. Ejtehadi, arxiv: [cond-mat.soft]) quantum information (L. Jin and Z. Song, arxiv: [quant-ph]) PT-symmetric quantum mechanics of fermions (K. Jones-Smith and H. Mathur, arxiv: [hep-th])

11 Backgrounds µ5 Ð*: µem î¹jü KUþŠ ½ E ê" þãïäl² šxd"3ù cƒm ù -< ÛÚ-<a,œ

12 Main content 1 Backgrounds Our model and some properties of pseudo-hermitian systems 3 Equivalence of our model and Pais-Uhlenbeck oscillator 4 Permutation and its spontaneous breaking in our model 5 The identity and its spontaneous breaking in the Pais-Uhlenbeck oscillator 6 Conclusion

13 Our model and some properties of pseudo-hermitian systems By adding a non-hermitian interacting term proportional to ip 1 p to the Hamiltonian of a free anisotropic planar oscillator we give a new Hamiltonian: H = 1 ( p 1 + p ) + 1 ( a 1 x 1 + a x ) + i a 3 a 1 a p 1 p, (1) where the non-vanishing constants a 1, a and a 3 are real, and a 1 a ; (x j, p j ), j = 1,, are Hermitian and satisfy: [x j, p k ] = iδ jk, [x j, x k ] = 0 = [p j, p k ], j, k = 1,. () If we apply the definition of operators P and T: P : x j x j, p j p j, i +i; T : x j +x j, p j p j, i i, (3)

14 Our model and some properties of pseudo-hermitian systems the Hamiltonian is PT-pseudo-Hermitian symmetry or PT-pseudo-Hermitian self-adjoint: H = H := (PT ) 1 H (PT ). (4) The property coincide with definition of the pseudo-hermitian symmetry: H = H := η 1 H η, (5) where η usually is required to be linear and Hermitian, but for an antilinear anti-hermitian η some fundamental properties for a consistent quantum system no longer exist. However, the antilinear anti-hermitian operator η = PT in our model is an special case and still complies with all the fundamental properties in quantum mechanics.

15 Our model and some properties of pseudo-hermitian systems 'u 5Ú ½Â 5½Âµ η(a ψ 1 + b ψ ) = aη ψ 1 + bη ψ. ve ^ 5ŽÎ 5 ψ 1 η ψ = ψ η ψ 1. ~X žm üžît 5 ŽÎ"

16 Our model and some properties of pseudo-hermitian systems We just list the main results as follows: A linear Hermitian η leads to a real probability, i.e. the real bilinear form of wavefunctions, ψ η ψ = ψ η ψ, also, our model associated with the anti-linear anti-hermitian PT does, i.e., ψ PT ψ = ψ PT ψ. A linear Hermitian η leads to the conservation of probability d with time, i.e. dt ψ η ψ = 0, such a conservation law is guaranteed with respect to the anti-linear anti-hermitian d η = PT, i.e., dt ψ PT ψ = 0. A linear Hermitian η leads to the unitarity of time evolution, i.e. ψ(t) η ψ(t) = ψ(0) η ψ(0), where ψ(t) = e iht ψ(0), also, such a unitarity is guaranteed with respect to the anti-linear anti-hermitian η = PT, i.e., ψ(t) PT ψ(t) = ψ(0) PT ψ(0).

17 Our model and some properties of pseudo-hermitian systems A linear Hermitian η leads to a real average value, A Av = A Av, for any physical observable A which satisfies A = A := η 1 A η. Quite interestingly, we proved our Hamiltonian model, though η = PT anti-linear anti-hermitian, H = 1 ( p 1 + p) 1 ( + a 1 x1 + ax ) a 3 + i p 1 p a 1 a has a real spectrum, which will be seen from the equivalence of our model and the Pais-Uhlenbeck oscillator. A linear Hermitian η leads to the η-pseudo-hermitian symmetry at any time: A (t) = A(t), for any physical observable A at the initial time, where A(t) = e +iht A(0)e iht, still, such a symmetry is guaranteed with respect to the anti-linear anti-hermitian η = PT.

18 Our model and some properties of pseudo-hermitian systems A linear Hermitian η leads to the usual equation of motion for the average value of A d (t), dt A (t) Av = i [H, A (t)] Av, if η and H do not explicitly contain time. This equation is still satisfied with respect to the anti-linear anti-hermitian η = PT. For the details, see the Appendix of our paper: Phys. Rev. A 85 (01) [arxiv: [quant-ph]].

19 Main content 1 Backgrounds Our model and some properties of pseudo-hermitian systems 3 Equivalence of our model and Pais-Uhlenbeck oscillator 4 Permutation and its spontaneous breaking in our model 5 The identity and its spontaneous breaking in the Pais-Uhlenbeck oscillator 6 Conclusion

20 Equivalence of our model and Pais-Uhlenbeck oscillator Pais-Uhlenbeck oscillator {0µžmêo$Ä d 4 x dt 4 + ( ω1 + ω ) d x dt + ω 1ωx = 0, Ù ω 1 Úω ê L«ªÇ" ë A. Pais and G.E. Uhlenbeck, Phys. Rev. 79, 145 (1950)"

21 Equivalence of our model and Pais-Uhlenbeck oscillator In order to establish the relationship between our mode and the Pais-Uhlenbeck oscillator ò JÑ!äkPT é 5 M î H = 1 ( p 1 + p) 1 ( + a 1 x1 + ax ) a 3 + i p 1 p a 1 a \Hamilton equations µ a 3 ẋ 1 = p 1 + i p, a 1 a a 3 ẋ = p + i p 1, a 1 a ṗ 1 = a 1x 1, ṗ = a x. (6) Eliminating the momenta p j and one of the coordinates x j respectively in eq. (6), we finally achieve the desired equation of motion, d 4 x j dt 4 + ( a 1 + a ) d ( ) x j dt + a1a + a 3 x j = 0, j = 1,. (7) 4

22 Equivalence of our model and Pais-Uhlenbeck oscillator Eq. (7) looks very much like the equation of motion of the Pais-Uhlenbeck oscillator d 4 x dt 4 + ( ω1 + ω ) d x dt + ω 1ωx = 0, but contains one more parameter a 3. It will be seen that our model includes the Pais-Uhlenbeck oscillator as a special case. We introduce two new parameters ω 1 and ω defined as follows: ω 1 + ω := a 1 + a, ω 1ω := a 1a + a 3 4. (8) It is obvious that the new parameters are just the frequencies in the Pais-Uhlenbeck oscillator if they have real solutions of the above quadratic algebraic equation.

23 Equivalence of our model and Pais-Uhlenbeck oscillator Solving eq. (8), we get the following three different cases. Case I: a 3 < a 1 a, (9) the two solutions are real and unequal, ω1 = 1 [ ω = 1 a 1 + a ± (a 1 a ) a 3 ], [ ] (a a1 + a 1 ) a a 3. (10) For this case our model is equivalent to the Pais-Uhlenbeck oscillator with unequal frequencies (A. Pais and G.E. Uhlenbeck, Phys. Rev. 79, 145 (1950); C.M. Bender and P.D. Mannheim, Phys. Rev. Lett. 100 (008) 11040).

24 Equivalence of our model and Pais-Uhlenbeck oscillator Case II: a 3 = a 1 a, (11) the two solutions are real and equal, ω 1 = ω = 1 ( a 1 + a ). (1) For this case our model is equivalent to the Pais-Uhlenbeck oscillator with the equal frequency (A. Pais and G.E. Uhlenbeck, Phys. Rev. 79, 145 (1950); C.M. Bender and P.D. Mannheim, Phys. Rev. D 78, 050 (008)).

25 Equivalence of our model and Pais-Uhlenbeck oscillator Case III: a 3 > a 1 a, (13) the two solutions are a pair of complex conjugate numbers, ω1 = 1 [ a1 + a ± i a3 ( ] a1 ) a, ω = 1 [ a1 + a i a3 ( ] a1 ) a. (14) For this case our model is beyond the region of the Pais-Uhlenbeck oscillator where ω 1 and ω must be real and positive. It is interesting to investigate whether our model has real spectra in this case.

26 Equivalence of our model and Pais-Uhlenbeck oscillator As a whole, our model covers the Pais-Uhlenbeck oscillator. the Pais-Uhlenbeck oscillator can also be depicted by a PT -pseudo-hermitian quantum system H = 1 ( p 1 + p) 1 ( + a 1 x1 + ax ) a 3 + i p 1 p a 1 a though the Pais-Uhlenbeck oscillator was used to be described by a PT -symmetric quantum system H = p γ iqx + γ (ω 1 + ω )x + γ ω 1ω y, Ù γ ~ê (p, x)ú(q, y) üéýcþ" ë C.M. Bender and P.D. Mannheim, Phys. Rev. Lett. 100 (008) 11040; Phys. Rev. D 78 (008) 050.

27 Main content 1 Backgrounds Our model and some properties of pseudo-hermitian systems 3 Equivalence of our model and Pais-Uhlenbeck oscillator 4 Permutation and its spontaneous breaking in our model 5 The identity and its spontaneous breaking in the Pais-Uhlenbeck oscillator 6 Conclusion

28 Permutation and spontaneous breaking in our model We first analyze in our model the permutation symmetry and its spontaneous breaking. The Hamiltonian H = 1 ( p 1 + p) 1 ( + a 1 x1 + ax ) a 3 + i p 1 p a 1 a is invariant under the following permutation transformation, a 1 a, a a 1 ; x 1 x, x x 1 ; p 1 p, p p 1. (15) In order to investigate the relation between the permutation symmetry and a real spectrum, we diagonalize the Hamiltonian. Through linear transformations four intermediate phase space variables, X 1, P 1, X, P, are adopt as follows:

29 Permutation and spontaneous breaking in our model X 1 = a 1x 1 α a x α α 1, P 1 = α 1 a 1 p a p ; X = a 1x 1 α 1 a x α 1 α, P = α a 1 p a p, (16) and then obtain the seemingly diagonalized form of the Hamiltonian: H = 1 U ω P α1 + 1 ( ) 1 + α 1 X U ω P 1 + α + 1 ( ) 1 + α X, (17) where two pure imaginary parameters α 1 and α and two real and positive parameters U and m are defined by:

30 Permutation and spontaneous breaking in our model α 1 := a 1 a + (a1 a ) a3, ia 3 α := a 1 a (a1 a ) a3, (18) ia 3 U := a a (a1 a ) a3, (19) a1 + a (a + 1 a ) a3 ω := a 4 1 a + a 3 /4, (0) there is a hidden relationship among the parameters α 1 and α : 1 + α 1 α = 0. (1)

31 Permutation and spontaneous breaking in our model Under the permutation eq. (15), the newly introduced parameters α 1, α, X 1, X, P 1, P transform as follows: α 1 α, α α 1 ; X 1 α 1 X 1, X α X ; P 1 1 α 1 P 1, P 1 α P. () We can verify that eq. (17) is also invariant under the transformation above.

32 Permutation and spontaneous breaking in our model Because (1 + α1 )/(1 + α ) = α 1 = 1/α < 0, the positivity of 1 + α1 or of 1 + α in eq. (17) is indefinite, but takes either of the two cases: Case (1) : 1 + α 1 > 0, 1 + α < 0, (3) Case () : 1 + α 1 < 0, 1 + α > 0. (4) The two cases are equivalent in dynamics.

33 Permutation and spontaneous breaking in our model In the eq. (17) case (1) or case () is definitely chosen for some given values of the three parameters a 1, a and a 3, which is different from that in the Hamiltonian eq. (1). That is the reason why we have to diagonalize Eq. (3). Let us rewrite eq. (17) for the two cases as follows: H (1), () = ±U 1 U 1 m 1 m ( ( U 1 α 1 P 1 ) U α P + 1 ) ( ) + 1 α1 mω U 1 X 1 mω ( ) α U X (5), where the upper and lower signs correspond to case (1) and case (), respectively, and the real and positive parameter m is defined by

34 Permutation and spontaneous breaking in our model m := ω (a 1 a ) /a 3 1. (6) Eq. (5) is a key step for us to change our model into a completely diagonalized form. We can see that the permutation symmetry is now spontaneously broken in eq. (5). Under the permutation transformation, case (1) and case () exchange to each other. It seems to be a difficulty that gives rise to the interchange between positive energy levels and negative ones. Fortunately, the spontaneous breaking of permutation symmetry makes the quantum system escape from this obstacle, that is, the permutation symmetry is now spontaneously broken in the individual H (1) or H () and no exchange between them will occur.

35 Permutation and spontaneous breaking in our model é 5»" HiggsÅ 'µ We may have some similarity if we compare the spontaneous breaking of permutation symmetry in our case with the spontaneous breaking of vacuum symmetry in the Higgs mechanism of gauge field theory. That is to say, the former chooses one branch of the PT -pseudo-hermitian Hamiltonian eq. (5), H (1) or H (), for giving a real spectrum free of the interchange of positive and negative levels while the latter one branch of vacuum states for producing massive gauge bosons.

36 Permutation and spontaneous breaking in our model Now we introduce the final phase space variables, X 1, P 1, X, P, α1 X 1 = U 1 X 1, U P 1 = 1 α 1 P 1; α X = U X, U P = α P, (7) and rewrite eq. (5) with the form of the standard formulation of a harmonic oscillator Hamiltonian, ( P H (1), () = ±U 1 1 m + 1 ) ( P mω X1 U m + 1 ) mω X. (8) It can be proved that the final phase space variables, though non-hermitian, X j X j and P j P j, satisfy the same Heisenberg commutation relations as the Hermitian phase space variables (x j, p j ),

37 Permutation and spontaneous breaking in our model [X j, P k ] = iδ jk, [X j, X k ] = 0 = [P j, P k ], j, k = 1,. (9) Naively, the energy spectrum of our model in Case I ( a 3 < a1 a ) seems to be ( E (1), () = ± n ) ( U 1 ω n + 1 ) Uω, n 1, n = 0, 1,,. (30) This case is equivalent to the unequal-frequency Pais-Uhlenbeck oscillator. Either case (1) or case () the negative eigenvalues appear in E (1) or E (), which would lead to the appearance of negative norms or ghost states in this fourth-order derivative model. It was regarded for a long time as a puzzling problem for the Pais-Uhlenbeck oscillator.

38 Permutation and spontaneous breaking in our model However, this problem has been solved recently first by the imaginary-scaling scheme (C.M. Bender and P.D. Mannheim, Phys. Rev. Lett. 100 (008) 11040; Phys. Rev. D 78 (008) 050) and then alternatively by the modified imaginary-scaling scheme or its equivalent indefinite-metric scheme (A.V. Smilga, SIGMA 5 (009) 017; A. Mostafazadeh, Phys. Lett. A 375 (010) 93; Phys. Rev. D 84 (011) ). By applying the latter to our model in Case I ( a 3 < a1 a ) we obtain the desired spectrum, E = ( n ) U 1 ω + ( n + 1 ) Uω, n 1, n = 0, 1,,. (31) We mention that the spectrum is independent of whether case (1) or case () is taken.

39 Permutation and spontaneous breaking in our model In this section, We obtain the real spectrum that is free of negative values or bounded below, and the spectrum is independent of the values taken for the three non-vanishing parameters a 1, a and a 3 constrained by the inequality a 3 < a 1 a. This outcome is reasonable because the original formulation of our Hamiltonian eq. (1) under the constraint a 3 < a 1 a should have a unique spectrum. As to the spontaneous breaking of permutation symmetry which has played a crucial role in giving a real spectrum free of interchange between positive and negative eigenvalues.

40 Main content 1 Backgrounds Our model and some properties of pseudo-hermitian systems 3 Equivalence of our model and Pais-Uhlenbeck oscillator 4 Permutation and its spontaneous breaking in our model 5 The identity and its spontaneous breaking in the Pais-Uhlenbeck oscillator 6 Conclusion

41 The identity and its spontaneous breaking in the Pais-Uhlenbeck oscillator Now we turn to the connection of our model and the Pais-Uhlenbeck oscillator. For the unequal-frequency case, ω 1 ω, the equation of motion eq. (8) is invariant under the permutation: ω 1 ω, ω ω 1. One cannot distinguish one of the two frequencies is larger or smaller than the other. This property is called identity of the two frequencies in attribute. However, one of the signs in Eq. (10) must be chosen as a prerequisite in order to calculate the spectrum of the unequal-frequency Pais-Uhlenbeck oscillator, In other words, the identity of the unequal ω 1 and ω must be broken spontaneously. For the choice ω 1 > ω, we get the relation of the parameters U and ω in our model and the frequencies ω 1 and ω in the Pais-Uhlenbeck oscillator,

42 The identity and its spontaneous breaking in the Pais-Uhlenbeck oscillator (a a ω 1 = U 1 1 ω = + a + 1 a) a 3, (3) (a a1 ω = Uω = + a 1 a) a 3. (33) Therefore, the spectrum of our model can be rewritten as ( E = n ) ( ω 1 + n + 1 ) ω, n 1, n = 0, 1,,, (34) which is just the formulation given in refs. (C.M. Bender and P.D. Mannheim, Phys. Rev. Lett. 100 (008) 11040; Phys. Rev. D 78 (008) 050; A.V. Smilga, SIGMA 5 (009) 017; A. Mostafazadeh, Phys. Lett. A 375 (010) 93; Phys. Rev. D 84 (011) ).

43 Main content 1 Backgrounds Our model and some properties of pseudo-hermitian systems 3 Equivalence of our model and Pais-Uhlenbeck oscillator 4 Permutation and its spontaneous breaking in our model 5 The identity and its spontaneous breaking in the Pais-Uhlenbeck oscillator 6 Conclusion

44 Conclusion We construct a concrete PT -pseudo-hermitian symmetry model to describe the Pais-Uhlenbeck oscillator. The model covers the Pais-Uhlenbeck oscillator as a special case. We proved that the fundamental properties for a consistent quantum system are satisfied by the anti-linear anti-hermitian operator η = PT. Our model is invariant under the permutation of two dimensions and such an invariance (symmetry) should be broken spontaneously in order to obtain a positive real spectrum. The permutation of two dimensions in our model corresponds to the identity of two frequencies in the unequal-frequency Pais-Uhlenbeck oscillator. For case III in our model, a 3 > a1 a, it is beyond the region of the Pais-Uhlenbeck oscillator and leaves it for further consideration.

45 Thank you!