$q$-Deformed Quantum Mechanics and the Thermodynamics of Black Hole/White Hole Spectral pair

q -Deformed Quan tum Mec hanics and t he Thermo dynamics of Blac k Hole/W hite Hole Sp ectral pair S. Jalalzadeh 1 , 2 , 3 ‡ , R. Jalalzadeh 4 , and H. Moradpour 4 1 Izmir Institute of T echnology , Department of Physics, Urla, 35430 , Izmir, T¨ urkiye 2 Cent er for Theo retical Physics, Khaz a r Universit y , 41 Mehseti Street, Baku, AZ1096, Azerbaijan 3 Department of Physics, Dogus University , Dudullu- ¨ Umraniye, 347 75 Is tanbu l, T¨ urkiye 4 Research Institute for Astronomy a nd Astr ophysics of Mara gha (RIAAM), Univ er sity of Maragheh, P . O . Box 5513 6-553 , Maragheh, Ir an E-mail: shahram jalal zadeh@iyte.edu.tr Abstract. In this work, we inv estigate the thermo dynamics of Sch warzsc hild black and white holes within a q -deformed Wheeler– DeWitt framework. By introducing a q -deformed Heisenberg–W eyl alg e bra at a ro ot of unity , we derive a finite-dimensional Hilber t space, a bo unded m as s spectr um, and an adiabatic inv ariant leading to a bo unded entropy-mass r elation. The defo rmation r esults in a universal log arithmic correctio n, as well as a minimum temp era ture and a ma ximum entropy that matches the de Sitter b ound. Also, we exa mine the int er pretation of a cold remnant, which is dynamica lly stable be cause its radia tion ra te appr o aches zero, even though its heat capa city remains negative. W e als o explore the hologr aphic implica tions of this limited entrop y . Our res ults thu s provide a consistent semiclassic al picture, where quantum deformatio n naturally in tro duce s an entrop y b ound, av oids divergences at the final ev ap ora tion stage, and suggests a smo oth tr a nsition fr om qua n tum gravity to cosmolog y . 1. In tro duction Blac k holes (BH) ha v e made significan t contributions to adv ancemen ts in theoretical ph ysics. M oreov er, in recen t decades, n umerous studies hav e rev ealed the presence of massiv e and sup ermassiv e BHs in the cen ters of t ypical galaxies. The recen t observ atio ns of gravitational w a v es b y the LIGO and VIRGO collab orations [1, 2, 3] from the merger of binary a stroph ysical BHs hav e unequiv o cally confirmed the existence of BHs in the univ erse. In t heoretical ph ysics, BH phenomena are crucial for comprehending quantum gra vity , a sub ject that has b een widely inv estigated ov er the last sev eral decades. ‡ Author to who m a ny corresp ondence should b e address ed q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 2 Lik e atoms, BHs can b e assigned discrete quantum lev els in semiclass ical and minisuperspace approac hes; unlik e atoms, ho w ev er, BHs radiate thermally through Ha wking emission. One of the primary go als of the quantum theory of BHs is to predict their emission sp ectra. The radiation sp ectrum, in t urn, is determined b y the sp ectra of BHs’ classical observ ables. When a ho le emits a quantum of BH radia t ion, it transitions from one classic al observ able eigenstate t o the next. In a though t exp erimen t, Bek enstein conducted a detailed calculatio n of the smallest increase in the horizon area resulting from the absorption of a particle b y a Kerr–Newman BH [4]. He has conv incingly shown that this sp ecific rise in the horizon area r epresen ts a constant v alue, stayin g inv a r ian t and unaffected b y the c haracteristics that define the BH, namely it s mass, c harge, a nd angular momen tum. In 1974, Bek enstein [5] prop osed, using the Bohr–Sommerfeld quan tization condition, that the horizon a rea of a Kerr– Newman BH is quantize d, with the horizon area in Planc k ar ea units b eing prop o r tional to an in teger. The quan tizatio n of the BH has b een the fo cus of a lot of work [6, 7, 8, 9, 10, 1 1, 12, 13, 14, 15]. V az and Witten [16] determined the mass eigenstate and eigenv alue of a Sc h w arzsc hild BH b y solving the Wheeler–DeWitt (WDW) equation within the framew ork of canonical quan tum gr avit y . The authors of [17] analyzed the effects of the generalized uncertain ty principle (G UP) on canonical quan tum gra vit y of BHs. Als o, the thermo dynamics of fractional-fracta l BHs is studied in the con text of the fractiona l WD W equation in Refs. [18, 19]. Most of these articles supp ort the idea that the en tropy of BHs is quan tized, i.e., S BH = γ n, n is a large inte ger n um b er , (1) where γ is a pure n um b er of order one. This s hows that the entrop y spectrum is quan tized and equidistan t fo r a spherically symmetric static BH, i.e., ∆ S BH = γ . Historically , quan tum groups hav e arisen from in v estigations into quantum in tegrable mo dels, emplo ying quan tum inv erse scattering tec hniques, whic h resulted in the deformation of classical matrix gro ups and their asso ciated structures. Lie algebras as discussed in the w orks of Kulish et al. [20], Skly anin [21], and F addeev [22]. It has b een established that quan tum groups assume a significan t role in v arious domains, including quan tum in tegrable systems [23], conforma l field theory [24], knot theory [25], solv able lattice mo dels [26], top ological quan tum computations [27], molecular sp ectroscop y [28], quan tum gravit y [29, 30, 31, 32, 33, 34], and quan tum cosmology [35, 36]. In the article [35], the autho r used the q -deformed Heisen b erg– W eyl algebra to find the entrop y of a Sc h w arzsc hild BH. Briefly , they used the relation b etw een t he ev en t horizon of a hole, A = 16 π M 2 G 2 , the relat io n o f the en tropy of the BH with the horizon area, S BH = A/ (4 G ), with the q -deformed mass sp ectrum of the BH to obtain the sp ectrum of the entrop y . They o btained a new q -deformed en tropy giv en b y the relation S q = π sin( π N ( n + 1 2 )) / sin( π 2 N ), where n is an integer a nd N is the q - deformation parameter. Although this relation reduces to Eq. (1) for the c-n um b er limit, i.e., N → ∞ , it is clear that it is not consisten t with relation (1). In this a rticle, w e sho w that the meticulous examination of calculations leads us t o a similar expression q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 3 as (1) f o r q -deformed BHs. In addition, w e obtained the temp erature and heat capacity of the resulting q -deformed pair. The structure of our pap er is outlined as follo ws. The next section is a concise summary of the q -deformed WDW equation related to a Sc h w arzsc hild blac k hole-white hole (WH) pair. In section 3, w e rederiv e the en tropy , temp erature, and heat capacit y p ertinen t to the mo del. The final section presen ts the k ey findings of our study . Note that in o ur w o rk, the ‘BH/WH pair’ denotes the tw o monotonic branc hes of a single finite q - deformed mass sp ectrum, rather than a literal astrophy sical binary system. 2. q -deformed Sch warzsc hild quan tum blac k hole Birkhoff ’s theorem implies that the Sc h warzs child mass is the only diffeomorphism- in v aria nt parameter o f t he v acuum solution. After reducing the spherically symmetric gra vitational system to its phys ical minisup erspace degree of freedom, the WD W equation can b e mapp ed to a one-dimensional harmonic-oscillator eigen v alue problem. In this represen tat ion, x denotes the dimensionless canonical minisup erspace v ariable of the reduced Sc hw arzsc hild geometry , while the corr esp onding eigenv alue is prop ortional to M 2 . F or completeness, a brief deriv atio n o f this equation is pro vided in App endix A. The WDW equation of Sch warzs child geometry can b e expressed a s [34]: − 1 2 d 2 ψ ( x ) d x 2 + 1 2 x 2 ψ ( x ) = 2 M 2 m 2 P ψ ( x ) , (2) where ψ ( x ) is the w a ve function of the BH, a nd m P = 1 / √ G is the Planck mass in natural units, i.e., ~ = c = k B = 1. Up on the use of ( 2 ), the mass sp ectrum M n is M n = m P √ 2 r n + 1 2 , n = 0 , 1 , 2 , ... . (3) T o effectiv ely dev elop the quan tum deformation of the ab ov e WD W equation, follo wing Ref. [34], we utilize the Heisen b erg–W eyl algebra link ed to the structure of the non-semisimple Lie algebra h 4 , whic h includes four generators { a + , a − , N , 1 } , where 1 is the m ultiplicativ e identit y , N = a † a is the n umber o p erator, and a − , a + are the usual annihilation and creation op erato r s with usual comm utation relations in quan tum mec hanics. Consequen tly , the WDW equation (2) turns t o : ( a + a − + a − a + ) | n i = 2 M 2 m 2 P | n i , (4) where | n i is the mass eigenstate, and ψ n ( x ) = h x | n i . The quantum Heisen b erg–W eyl algebra, U q ( h 4 ), at the ro ot of unity , is a q - deformation of the Heisen b erg–W eyl algebra, whic h is a asso ciativ e unital [37, 38] C ( q )- algebra with generator s { a + , a − , q ± N/ 2 } . Note that in the undeformed oscillator represen ta tion, the op erato r a † a acts as the usual num b er o p erator on the F o c k basis. In the q -deformed algebra, how ev er, N is in tro duced as an indep enden t generator satisfying q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 4 the defining comm utatio n relations of the deformed Heisen b erg–W eyl algebra. Although it reduces to the usual n um b er op erato r role in the undeformed limit, it should not b e iden tified a priori with t he op erator a † a [37]. The generators s atisfy the follo wing quan tum defor med ( q -defo r med) comm utation relations [37] a − a + − q 1 2 a + a − = q N 2 , [ N , a ± ] = ± a ± , a † ± = a ∓ , N † = N , | q | = 1 , ( 5 ) where q is a primitiv e ro ot of unity , i.e., q := exp  2 π i N  , N is a na t ur a l n um b er, and N ≥ 2. The natural length scale in quantum grav ity is the Planck length. Therefore, N can b e expresse d as a function of the gravitational constant G o r the square of t he Planc k length: N = N ( l 2 P ). In the classical gravit y limit, where l P approac hes zero, N tends to infinity , indicating a transition to classical gravit y . This sho ws the q -deformation is a quan tum gravit y effect, with N ∝ 1 /l 2 P . T o maintain N as a dimensionless parameter, ano ther length scale is needed, where the ratio o f the Planck length t o this new length defines t he q - deformation parameter: N = L 2 q /l 2 P . The par a meter L q ma y b e in terpreted as an infrared length scale asso ciated with the finite-dimensional q -deformed represen ta tion. At the algebraic lev el, the k ey input is t he finiteness of the represen tation dimension N . The iden tification N = L 2 q /l 2 P is an additional ph ysical parametrization that allo ws one to asso ciate the finite sp ectrum with a la rge-scale cutoff. The classical gra vity limit is reac hed by setting L q → ∞ . Also, this deformation para meter pro vides a holographic view o f q uantum mec hanics [39 ]. Sp ecifically , the Hilb ert space of q - deformed neutral hy drogen gas in de Sitter space satisfies the strong holographic b ound with this parameter. One can easily sho w that the first t w o relations (5) a re actually equiv alent to the follo wing r elat io ns a + a − = [ N ] , a − a + = [ N + 1] , (6) where [ x ] := q x 2 − q − x 2 q 1 2 − q − 1 2 = sin  π x N  sin  π N  . (7) The q -deformed creatio n and annihilation op erators act on the mass states b y a + | n i = p [ n + 1] | n + 1 i , a − | n i = p [ n ] | n − 1 i , N | n i = n | n i , a + | N i = 0 , n = 0 , 1 , ..., N − 1 . (8) No w, like the ordinary F o c k space of the harmonic oscillator, w e can construct the represen ta tion o f the U q ( h 4 ) in the q -deformed F o c k space spanned by normalized eigenstates | n i | n i = 1 p [ n ]! a n + | 0 i , n = 0 , 1 , ..., N − 1 , (9) where the q-factoria l is defined b y [ n ]! := Q n m =1 [ m ]. The op erato r 2 M 2 m 2 P = a + a − + a − a + = [ N + 1] + [ N ] , (10) q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 5 is the q -analogue of the mass op erator defined in (4). Consequen tly , the ab o v e quan tum deformat io n o f the Sc h warzs ch ild geometry give s us the followin g eigen v alues for the q -deformed ma ss [34] M n = m P 2 s sin( π N ( n + 1 2 )) sin( π 2 N ) , n = 0 , ..., N − 1 . (11) Note that for N → ∞ the earlier eigen v alues will reduce to (3). Also, there is a tw o-fo ld degeneracy at the mass eigen v alues. The ma ss of the ground state n = 0, a s w ell as the state n = N − 1, are M 0 = M N − 1 = m P 2 . These sho w that the g round state mass is not deformed, and its v alue is the same as t he non-defo r med spectrum obtained in (3). Besides, the sp ectrum is b ounded in whic h the most excited state, n = N − 1 , has the same mass and a r ea as the ground state. Let us assume N is an ev en n um b er. In the finite sp ectrum, the sector 0 ≤ n ≤ N 2 − 1 is iden tified as the BH branch b ecause the mass incre ases with n , so do wn w ard transitions n → n − 1 corresp ond to mass lo ss, as exp ected for Hawking emission. The sector N 2 ≤ n ≤ N − 1 is corresp ondingly interprete d as a WH branc h, since the same b ounded sp ectrum is tra ve rsed b ey ond the maximal-ma ss states a nd dM n dn < 0. W e stress that ‘BH’ and ‘WH’ here refer to the t wo monotonic branc hes of a single minisup erspace sp ectrum, not to a literal tw o -b o dy astroph ysical system [34]. States with maximum mass eigen v alue M max = m P 2 p cot( π 2 N ) are n = N / 2 − 1 and n = N / 2. T o illustrate t his p oin t, Fig. 1a sho ws the mass sp ectrum of a q -deformed BH with N = 30. Using (11) one c hec ks ∂ M n ∂ n ∝ cos  π N ( n + 1 2 )  , hence M n increases for 0 ≤ n < N / 2 (BH bra nc h) and decreases for n > N / 2 (WH branc h). The stationary p oin ts at n = N / 2 − 1 , N / 2 are maxima and yield M max = m P 2 p cot( π 2 N ), confirming the t w ofo ld degeneracy . Note that dM /dn > 0 for n < N / 2 and < 0 for n > N / 2. In additio n, the w a ve functions are q-Hermite p olynomials whose prop erties can b e found in Refs. [36, 4 0]. 3. Thermo dynamics of q -deformed black and white hole pair W e assume Ha wking radiation from a massiv e BH (WH) emitted during a transition from state n to state n − 1 (fr o m state n − 1 to state n ). Also, we consider that M ≫ m P , n ≫ 1, and N ≫ 2, whic h implies that sin( π 2 N ) = tan( π 2 N ) ≃ π 2 N . It has b een established [41, 42] that the dynamics of the BH o ccur b etw een discrete mass eigenstates giv en by Eq. ( 11). Then, using the q -deformed mass sp ectrum (1 1) giv es us the frequency o f emitted (or absorb ed) radiation [19]. F or adjacent levels , | ∆ M | = | M n − M n − 1 | ≃   ∂ M ∂ n   (with ∆ n = 1). Differentiating (11) w.r.t. n a nd expanding for large N yields ∂ M ∂ n ≃ m P 8 M n h 1 + m 2 P 8 M 2 n − 1 2  2 π M 2 n N m 2 P  2 i . Using ω = | ∆ M | 2 π and t P = 1 /m P ensures dimensional consistency in natural units, ω = | ∆ M | 2 π ≃ m P r 1 −  2 π M 2 n N m 2 P  2 8 π M n t P   1 + m 2 P 8 M 2 n s 1 −  2 π M 2 n N m 2 P  2   , (12) q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 6 (a) Th e mass sp ectrum of a q - deformed BH-WH p air. (b) The entrop y sp ectrum of a q - deformed BH-WH pair. Figure 1: Fig. (a) shows the mass sp ectrum of a q -deformed BH-WH with N = 30 . Fig. (b) illustrates the entrop y of the same BH- WH pair. where | ∆ M | = | M n − M n − 1 | , and t P is t he Planck time. Not e that at the limit N → ∞ , the ab ov e relatio n reduces to ω ≃ m P 8 π M n t P 1 + 1 8  m P M n  2 ! , n ≫ 1 , (13) whic h is the frequency of the radiatio n obt a ined in ordinary BHs [1 9]. The follo wing adiabat ic inv ar ia n t of the mo del, a s demonstrated in R efs. [19, 43 , 44] for ordinary BHs, gives the en t r o p y o f a q -deformed BH (WH): S ( q ) = Z M m P / 2 d M ω ≃ 2 N arcsin  S BH 2 N  − π 2 ln ( S BH ) + const. , (14) where S BH = 4 π M 2 /m 2 P is the Bek enstein–Ha wking en tropy of an ordinary Sc h w arzsc hild BH. A brief deriv atio n o f Eq. (14), sho wing explicitly how the adiaba t ic- in v aria nt integral yields bo th the arcsine contribution and the lo g arithmic correction, is presen ted in App endix B. Also, b y assuming S BH 2 N ≪ 1, and using the T a ylor series expansion of a rcsin( x ), the en tropy (14) give s us the cubic correction to the hole en tropy: S ( q ) ≃ S BH + 1 24 N 2 ( S BH ) 3 − π 2 ln ( S BH ) . (15) Note that for N → ∞ , the aforemen tioned q -deformed entrop y con v erges to the Ha wking–Bek enstein en trop y with a logarithmic correction, where the co efficien t − π / 2 arises fro m the mo dified state densit y in the q -deformed measure, distinguishing it fro m the LQG v a lue − 3 / 2. term: S = S BH − π 2 ln ( S BH ) + const. (16) q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 7 Therefore, the q -deformatio n generates a negat iv e loga r it hmic correction to the en tropy , in addition t o the b ounded arcsine con tribution asso ciated with the finite-dimensional sp ectrum. It is w orth fo cusing on the logarit hmic term obtained here (ln( S B H )). Previous studies insist on the univ ersalit y of this term [45, 46, 47, 4 8, 49 , 50, 5 1, 52, 53, 54, 5 5, 56, 57]. Therefore, the result indicates that just lik e the framew orks of quan tum geometry of the horizon [45 , 46, 47, 48 , 49, 50, 5 1, 52] and thermo dynamic fluctuatio ns around the equilibrium [5 3, 54 , 55, 56, 58, 59, 60, 57], the q - deformed algebra also requires this term. F urthermore, comparing the differen t ia l of (14) with the first law of thermo dynamics for BHs, d M = T H d S , prov ides the temp erature o f t he q -deformed BH-WH pair: T ( q ) = m P 8 π M s 1 −  2 π M 2 N m 2 P  2    1 + m 2 P 8 M 2 s 1 −  2 π M 2 N m 2 P  2    T P , (17) where T P is the Planc k temp erature. It is easy to ve rify tha t the com bination of Eqs. (17) and (14), a nd ig no r ing the loga r it hmic term, giv es us M = 4 N T (q) tan  S (q) 2 N  . (18) This equation is the q -deformed Smarr’s fo rm ula for the mass of a BH. In the limit N → ∞ , tan( S ( q ) / (2 N )) → S BH / (2 N ) and the standard Smarr relation M = 2 T H S BH [61] is recov ered. It is clear that fo r N → ∞ , this relation reduces to Smarr’s formula M = 2 T H S BH . Also, the heat capacity is giv en b y C ( q ) = dM dT ( q ) = − 8 π M 2 r 1 −  2 π M 2 N m 2 P  2 m 2 P  1 +  2 π M 2 N m 2 P  2  . (19) One can easily find out t ha t the ab ov e t emp era t ur e and heat capacity reduce to the Ha wking temp erature, T H = m P / (8 π M ) T P , and the heat capacit y , C = − 8 π M 2 /m 2 P , of the Sc h warzs ch ild hole at N → ∞ limit. As in the Sch w arzsc hild case, Hawk ing emission driv es the BH branch tow a rd lo we r mass. The negativit y of the heat capacit y in Eq. (19) implies that this mass loss is accompanied b y an increase in temp erat ure along the branc h. On the other hand, the WH radiates to the insides of its horizon, and it gains mor e mass until its mass reac hes the maximum mass M max = m P 2 p cot( π 2 N ) ≃ m P p N / (2 π ). T o clarify these principles, w e shall examine m ultiple virtual particle pairs that arise in the vicinit y of the outer region of the horizon. The in tense grav itatio na l field a pplies a strong er att r a ctiv e for ce on the particle that is closer in pro ximity compared to the one that is fa rther aw ay; therefore, the gravitational pull of the BH generates a tidal force that acts to disso ciate the virtual pair. The substan tial force presen t near the eve nt horizon culminates in t he q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 8 irrev ersible separation o f particles, resulting in their con ve rsion into real part icles, a s they are no longer able to reunite fo r mutual annihilation. The particle that is closest to the horizon is engulfed, while the one that is more distant is allow ed to escap e to infinity due to the energy transferred by the tidal force. Consequen tly , there is a decrease in the mass of the BH. In con trast, the conditions surrounding a WH are o pp osed to the previously discussed situation. If w e consider the creation of virtual particles within the horizon o f a WH, the particle crosses the horizon and is unable to re-ente r the interior of the horizon. In the case of a WH, it gains mass through the pro cess of Hawk ing radiation. Note that inserting the ma ss sp ectrum (11) in to (14), and ignoring the logarithmic term, giv es us the discrete q -deformed en tropy of a massiv e BH-WH pair as S ( q ) = ( 2 π  n + 1 2  , 0 ≤ n ≤ N 2 − 1 , (BH) , 2 π  N − n − 1 2  , N 2 ≤ n ≤ N − 1 , (WH) . (20) This en tropy relation is consisten t with the Bek enstein prop osal [5], giv en b y Eq. (1). Ho w ev er, we should note that the q - deformed en tropy is bounded from ab ov e. T o illustrate this p oin t, F ig. 1b show s the en trop y sp ectrum of a q -deformed BH-WH pair for N = 30. The maxim um p ossible v alue for en tropy (for n = N / 2 − 1 or n = N / 2) is S (max) ( q ) = π ( N − 1) ≃ π  L q l P  2 . (21) Therefore, the WH g ains more mass via Haw king radiation and ac hiev es its maxim um en trop y giv en b y the ab ov e relation. Note that the existence of maxim um en tropy and also maxim um mass, giv en b y Eq. (11) a re direct results of q -deformation. In addition, the temp erature and the heat capacit y o f t his state are giv en by T (min) ( q ) ≃ r π 128  l P L q  3 , C (max) (q) ≃ − π . (22) Eqs. (21) and (2 2) demonstrate that the ultimate state of WHs is univ ersally applicable, irresp ectiv e of the initial mass of the WH, and is wholly a manifestation of quan tum gra vitational phenomena. W e emphasize t ha t the analysis of Sections 2–3 is p erformed fo r the q -deformed minisuperspace quantization of the Sch warzsc hild geometry , no t for a Sch warzs ch ild– de Sitter metric. The quan tity Λ q = 3 /L 2 q en ters only a s an effectiv e large-scale parameter inferred from the maximal en tro py S max q , whic h coincides forma lly with the de Sitter en tro py . Our a nalysis do es not quan tize a Sch warzs child–de Sitter geometry . Th us, the de Sitter relatio n S (max) ( q ) = 3 π / ( G Λ q ) should b e view ed as an effectiv e holographic/cosmological corresp ondence implied b y the finite-dimensional q - deformed Hilb ert space, ra ther than as an assumption ab out the asymptotics of the blac k-hole spacetime studied in Sections 2 and 3. Also, at the algebraic lev el, the key input is simply that q is a ro ot of unity , whic h mak es the represen ta t ion finite dimensional with q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 9 dimension N . All sp ectral b oundedness results in Eqs. (11), (21), a nd (22 ) follo w fro m finite N alone. The parametrization N = L 2 q /l 2 P is an additional phys ical iden tification that in tro duces a n infrar ed length scale L q ; only af t er this identific atio n do es one obtain the effectiv e cosmological quan tity Λ q = 3 /L 2 q . 4. Conclusions Quan tum groups provide more intricate symmetries t ha n con v en tional Lie algebras, of whic h they are a sp ecial subset. This suggests that quan tum groups ma y b e suitable for c haracterizing the symmetries of ph ysical sys tems that exceed the limita t io ns of Lie algebras. F urthermore, q -deformed mo dels pro vide a notable b enefit due to their asso ciated Hilbert space b eing finite-dimensional when q , the deformation para meter, is a ro ot of unit y [38]. This indicates that using quan tum g roups with a deformation parameter of the ro ot of unity is adv antageous for deve loping mo dels with a limited n um b er of states. These mo dels can b e used to in ve stigate applications in quantum gra vity and quantum cosmology that conform to the holographic principle and UV/IR mixing, addressing the cosmological constant problem [36]. In a more explicit manner, within the framew ork of cosmology , the quan tum deforma t io n para meter L q , whic h is articulated subsequen t to the comm utation relations (5 ), results in the emergence of a cos molog ical constan t represen ted as Λ q = 3 /L 2 q . Iden tifying N = L 2 q /l 2 P implies S max ( q ) = π ( L q /l P ) 2 = 3 π G Λ q , equal to the de Sitter en tropy for Λ q , th us linking the IR deformation scale to cosmology . In this w ork, w e studied the thermo dynamic implications of the q -deformed minisuperspace quantization of the Sc hw arzsc hild black hole. The ro ot-of-unity deformation leads to a finite-dimensional Hilb ert space and, consequen tly , t o a b ounded mass s p ectrum with t w o monotonic branches , whic h w e in terpret as blac k-hole and white-hole sectors of a single quan tum sp ectrum. Using the adiabatic-inv ar ia n t metho d, w e obtained a mo dified entrop y formula containing b oth a b ounded arcsine term and a logarithmic correction. The corresp onding temp erature and heat capacit y exhibit a minim um temp erature and a maximal entrop y , suggesting an effectiv e infrared cutoff scale. The relation of this cutoff to an effectiv e cosmological constan t was discussed at the thermo dynamic lev el. F urt her in v estigation is needed to determine whether these features p ersist b ey ond the presen t minisup erspace framew ork a nd whether they admit observ able phenomenological consequences . P oten tial, alb eit challenging, signatures include fain t gravitational-w a v e ech o es from transitions b et w een adjacen t q -lev els. Extending the ana lysis to Reissner–Nordstr¨ om and Kerr–Newman geometries is a prior it y f or astroph ysical relev ance. q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 10 App endix A. Canonical reduction of the Sc h wa rzschild blac k hole and t he oscillator form of the WD W equation In t his app endix, w e briefly review the cano nical reduction of the Sc h w arzsc hild BH that underlies the WDW equation used in the main text. Sinc e Eq. ( 2 ) play s a cen tral role in our analysis, it is useful to summarize how it a rises fr o m the reduced phase-space quan tization of spherically symmetric gravit y . W e b egin with the spherically symmetric ADM line elemen t ds 2 = − N ( r , t ) 2 dt 2 + Λ( r , t ) 2  dr + N r ( r , t ) dt  2 + R ( r , t ) 2 d Ω 2 , ( A.1 ) where d Ω 2 is the line elemen t on the unit t w o-sphere S 2 . W e adopt Kuc ha ˇ r’s fall-off conditions [62], whic h ensu re that the co ordinates r and t extend o ve r the Krusk al manifold, −∞ < r , t < ∞ , while the geometry remains asymptotically flat at b oth spatial infinities. These conditions also imply that the ADM 4-mo mentum has no spatial c omp onen t at r → ±∞ , so that the BH is at res t with respect to the left and right asymptotic Mink ows ki regio ns. Fixing the asymptotic v alues of the lapse function at the tw o infinities to b e time-dep enden t functions N ± ( t ), the Einstein–Hilb ert action, supplemen ted b y the appropriate b o undary terms, tak es t he Hamiltonian form S = Z dt Z ∞ −∞ n Π Λ ˙ Λ + Π R ˙ R − N H − N r H r o dr − Z n N + M + + N − M − o dt, (A.2) where the canonical momenta conjuga te to Λ and R are Π Λ = − m 2 P N R  ˙ R − R ′ N r  , Π R = − m 2 P N h Λ  ˙ R − R ′ N r  + R  ˙ Λ − (Λ N r ) ′  i . (A.3) Here m P = 1 / √ G is the Planc k mass, ˙ f = ∂ t f , and f ′ = ∂ r f . The quan tities M ± ( t ) are determined by the a symptotic fall-off of the canonical v ariables and, on classical solutions, coincide with the Sc h w arzsc hild mass. The sup er-Hamiltonian and radial sup er-momen tum constrain ts a re giv en b y H = − 1 Rm 2 P Π R Π Λ + 1 2 R 2 m 2 P Π 2 Λ + RR ′′ Λ − RR ′ Λ ′ Λ 2 + R ′ 2 2Λ − Λ 2 , H r = 1 m 2 P (Π R R ′ − Λ Π ′ Λ ) . (A.4) F ollowing Kuc ha ˇ r [62], one intro duces the canonical transformatio n from the q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 11 original v aria bles (Λ , Π Λ ; R, Π R ) to the new pairs ( M , Π M ) and ( R , Π R ), M = Π 2 Λ 2 m 4 P R − RR ′ 2 2Λ 2 + R 2 , Π M = ΛΠ Λ m 2 P "  R ′ Λ  2 − 1 m 4 P  Π Λ R  2 # − 1 , R = R, Π R =  Π Λ H m 2 P R + R ′ H r Λ 2  "  R ′ Λ  2 − 1 m 4 P  Π Λ R  2 # − 1 . (A.5) In these v a riables, the action b ecomes S = Z dt Z ∞ −∞ n ˙ M Π M + ˙ R Π R − N r H r − N H o dr − Z { M + N + − M − N − } dt, (A.6) where the transformed constraints tak e the fo rm H = −  1 − 2 M m 2 P R  − 1 M ′ R ′ + m − 4 P  1 − 2 M m 2 P R  Π M Π R   1 − 2 M m 2 P R  − 1 R ′ 2 − m − 4 P  1 − 2 M m 2 P R  Π 2 M  1 / 2 , H r = 1 m 2 P (Π M M ′ + Π R R ′ ) . (A.7) V ariation of the a ction with resp ect to N and N r imp oses the constraints H ≈ 0 , H r ≈ 0 , (A.8) whic h are equiv a lent to M ′ ≈ 0 , Π R ≈ 0 . (A.9) The first relation implies that M is indep enden t of the radial co ordinate, namely M = M ( t ). After imp osing Π R ≈ 0 and M = M ( t ), o ne obta ins a r educed action in terms of the single canonical pair ( M , P ), where P = Z ∞ −∞ Π M d r = − Z ∞ −∞ r  dR d r  2 − Λ  1 − 2 M m 2 P R  1 − 2 M m 2 P R d r , −∞ < P < ∞ . (A.1 0 ) The reduced action then ta k es the fo r m S = Z n P ˙ M − ( N + + N − ) M o d t. (A.11) The v ariables ( M , P ) satisfy the canonical P oisson brack et { M , P } = 1. Cho osing, as in Ref. [63], the asymptotic Mink ows ki time on the right-hand side as the ev olution parameter amounts to fixing N + = 1 , N − = 0 , (A.12) q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 12 so that the reduced a ctio n b ecomes S = Z n P ˙ M − H ( M ) o d t, (A.13) with reduced Hamiltonian H ( M ) = M . (A.14) The corresponding equations of motion yield M = const. and P = − t , in agreemen t with Birkhoff ’s theorem: the Sc h w arzsc hild mass is the only gauge-inv a rian t constant of motion of the v acuum solution. The conjugate momen tum P ma y therefore be in terpreted as the asymptotic time separation asso ciated with the spatial h yp ersurface. T o pro ceed tow a r d quan tization, one follo ws the Euclidean argumen t of Ref. [63]. Since the Euclidean Sc hw arzsc hild geometry is regular only if the Euclidean time is p erio dic with p erio d T − 1 H , where T H = 1 8 π GM , (A.15) one imp oses the iden t ificatio n P ∼ P + 1 T H . (A.16) This p erio dicit y remov es the conical singularity at the hor izon in the Euclidean section. A t the same time, it sho ws that the ph ysical phase space is no t the full ( M , P )-plane, but rather a wed ge b ounded by the M - axis and the line P = T − 1 H . It is therefore conv enien t to in tro duce a new canonical pair ( x, p ) that un wraps this w edge-shap ed phase space and incorp orates t he p erio dic iden tification naturally: x = 2 √ G M cos(2 π T H P ) , p = 2 √ G M sin(2 π T H P ) . (A.17) Using { M , P } = 1, one readily v erifies that this transformation is canonical, { x, p } = 1. Moreo v er, squaring and adding t he t w o relatio ns in Eq. (A.17) gives M 2 = m 2 P 4  x 2 + p 2  . (A.18) Th us, in t he reduced phase space, the Sch warzs ch ild mass is mapp ed to the Hamiltonian of a one-dimensional harmonic oscillator. Up on cano nical quan tization in the x - represen ta tion, p → − i d d x , (A.19) one obtains  − 1 2 d 2 d x 2 + 1 2 x 2  ψ ( x ) = 2 M 2 m 2 P ψ ( x ) , (A.20) whic h is precisely Eq. (2) in the main text. W e therefore in terpret x as the dimensionless canonical minisup erspace v ariable of the reduced Sc h w arzsc hild BH, in terms o f whic h the WD W equation takes the form of a harmonic-oscillator eigenv alue problem. q -De forme d Quantum Me chanics and the Thermo dynamics of B lack Hole /White Hole Sp e ctr al p a ir 13 App endix B. Deriv ation of the en tropy form ula In this app endix, w e provide an interme diate deriv a t io n of Eq. (14), sho wing explicitly ho w the adia batic-in v a r ia n t in tegral yields b oth the arcsine term and the logarithmic correction. Starting from Eq. (12), w e treat the mass sp ectrum semiclassically and regard M n as a contin uous v ariable M . The inv erse of the transition frequency can then b e written as 1 ω ( M ) = 8 π M m 2 P r 1 −  2 π M 2 N m 2 P  2     1 + m 2 P 8 M 2 r 1 −  2 π M 2 N m 2 P  2     − 1 ≃ − π M + 8 π M m 2 P r 1 −  2 π M 2 N m 2 P  2 , (B.1) where, in the second equalit y , w e did expand 1 /ω ( M ) up to the first subleading order. The corresp onding a dia batic in v a rian t is I = Z d M ω ( M ) ≃ Z 8 π M d M m 2 P r 1 −  2 π M 2 N m 2 P  2 − π Z d M M . (B.2) The first integral g ives Z 8 π M d M m 2 P r 1 −  2 π M 2 N m 2 P  2 = 2 N a rcsin  2 π M 2 N m 2 P  . (B.3) The second in tegral is elemen ta ry: − π Z d M M = − π ln M + const. (B.4) Using the Bek enstein–Ha wking entrop y S BH = 4 π M 2 m 2 P , (B.5) one has − π ln M = − π 2 ln S B H + const. 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