P. N. Lebedev Physical Institute, Leninsky Prospect 53, 117924 Moscow, Russia
이 논문에서는 Brownian motion을 q-deformations 한 방법에 대한 연구 결과를 제공합니다. q-deformed Brownian motion은 일반적인 Brownian motion과 유사하지만 cumulant structure와 non-Gaussian nature가 있습니다. 이 논문에서 우리는 q-deformation의 기초를 다루고, q-deformed Brownian motion의 특성을 정의하고, 이론을 확인하는 테스트를 수행합니다.
q-Deformation은 algebraic structure의 일반화이며, stochastic process의 영역으로 확장되었다. 이 연구는 물리학적 현상을 이해하는 새로운 방법을 제공하며, 더 나아가 q-deformation이란 어떤 물리적 현상에 적용할 수 있는 유용한 도구라는 것을 보여주고 있습니다.
q-Deformation의 특성:
1. Cumulant structure: q-deformed Brownian motion은 일반적인 Brownian motion과 다르며 cumulant structure와 non-Gaussian nature가 있습니다.
2. Non-Gaussian nature: q-deformed Brownian motion은 Gaussian process가 아니며 cumulants가 모든 orders에 대해 존재하고 order가 3 이상인 경우에만 0이 아닌 경우를 갖습니다.
q-Deformation의 기초:
1. q-deformed algebra: q-deformed algebra는 Lie 산술의 일반화이며, q-Deformation의 기초입니다.
2. Fock space: Fock 공간은 bosonic 자유 필드에 포함될 수 있는 Brownian motion을 의미합니다.
3. stochastic process: stochastic process는 random 시계열을 의미하며, 물리학적 현상에 적용할 수 있습니다.
q-deformed Brownian motion의 특성:
1. independent increments: q-deformed Brownian motion은 일반적인 Brownian motion과 유사하며 independent increments를 갖습니다.
2. martingale property: q-deformed Brownian motion은 일반적인 Brownian motion과 유사하며 martingale property를 갖습니다.
q-Deformation의 응용:
1. 물리학적 현상: q-Deformation은 물리학적 현상을 이해하는 새로운 방법을 제공하며, 더 나아가 q-deformation이란 어떤 물리적 현상에 적용할 수 있는 유용한 도구라는 것을 보여주고 있습니다.
2. 통계 기하학: q-Deformation은 통계 기하학에 응용될 수 있으며, random 시계열을 모델링하는 데 사용될 수 있습니다.
결론:
q-Deformation은 물리학적 현상을 이해하는 새로운 방법을 제공하며, 더 나아가 q-deformation이란 어떤 물리적 현상에 적용할 수 있는 유용한 도구라는 것을 보여주고 있습니다. 이 연구는 Brownian motion의 non-Gaussian nature와 cumulant structure를 다루며, q-deformed Brownian motion의 특성을 정의하고 테스트합니다.
P. N. Lebedev Physical Institute, Leninsky Prospect 53, 117924 Moscow, Russia
arXiv:hep-th/9303141v1 25 Mar 1993CERN-TH-6838/93q-Deformed Brownian MotionV. I. Man’koP.
N. Lebedev Physical Institute, Leninsky Prospect 53, 117924 Moscow, RussiaR. Vilela Mendes∗Theory Division, CERN, CH 1211 Geneva 23, SwitzerlandAbstractBrownian motion may be embedded in the Fock space of bosonic free fields in onedimension.
Extending this correspondence to a family of creation and annihilationoperators satisfying a q-deformed algebra, the notion of q-deformation is carriedfrom the algebra to the domain of stochastic processes. The properties of q-deformedBrownian motion, in particular its non-Gaussian nature and cumulant structure, areestablished.CERN-TH-6838/93March 1993∗Permanent/Mailing address: CFMC, Av.
Gama Pinto 2, 1699 Lisboa Codex, Portugal
The concept of symmetry plays an essential role in the description of physical phe-nomena. In most cases this symmetry is related to covariance under the transformationsinduced by a Lie algebra.
A generalization of this mathematical structure, the q-deformed(or quantum) algebras, has recently emerged[1−7]. q-deformed algebras, first discoveredin the context of integrable lattice models, were later identified as an underlying math-ematical structure in topological field theories[8] and rational conformal field theories[9].Other attempts to apply the notion of q-deformed algebras cover a wide range of differentdomains, from space-time symmetries[10−12] to gauge fields[13], to quantum chemistry [14].In view of the actual and potential applications of q-deformation in the context of Lie al-gebras and superalgebras, it is interesting to ask whether the notion of q-deformation canalso be extended to other (non-algebraic) mathematical structures.
In this paper we tryto extend this notion to stochastic processes. Our starting point is the well-known embed-ding of Brownian motion in the Fock space of bosonic free fields in one dimension[15,16].Extending this correspondence to a time family of creation and annihilation operatorssatisfying a q-deformed algebra we establish a q-deformation of Brownian motion.q-deformed creation and annihilation operators were defined by several authors[17−20].They satisfy the algebraaa† −q−1a†a = qN(1.a)aa† −qa†a = q−N(1.b)where N is the number operator[N, a†] = a†[N, a] = −a(2)The operators aa† may be realized as infinite-dimensional matrices on a vector space bya†|n >=q[n + 1]|n+1 >a|n >=q[n]|n−1 >N|n >= n|n > (3)where we used the notation[X] = Xq = sinh(X ln q)sinh(ln q)(4)X being a number or an operator.
q-deformation of single boson operators is invariantunder the replacement q →q−1 and we write the algebra in an explicitly symmetric formwhich will be useful later on.aa† −12(q + q−1)a†a = 12(qN + q−N)(5)(Notice that (q + q−1) = [2])We now consider a family { aτ , a†τ } of q-deformed operators labelled by a continuoustime parameter and a scalar fieldφτ = aτ + a†τ(6)1
For the { aτ , a†τ } family we generalize the relations (5) and (2) toaτ1a†τ2 −12(q + q−1)a†τ2aτ1 = 12(qN + q−N)δ(τ1 −τ2)(7)[N, a†τ] = a†τ[N, aτ] = −aτ(8)This is the simplest extension of the relations to a family of q-deformed operators labelledby a continuous parameter. Other generalizations of (5) are possible, involving for examplebraid relations at different times.
Notice also that, for our purposes of construction of astochastic process, no assumptions are needed concerning the commutation properties ofaτ1 aτ2 and a†τ1 a†τ2 at different times.Smearing the fields with characteristic functions χ[0,t] of the interval [0,t]aq(t) = aq(χ[0,t]) =tZ0dτaτ(9.a)a†q(t) = a†q(χ[0,t]) =tZ0dτa†τ(9.b)φq(t) = φq(χ[0,t]) =tZ0dτφτ(10)the algebraic relations becomeaq(t1)a†q(t2) −12(q + q−1)a†q(t2)aq(t1) = 12(qN + q−N) < τ1|τ2 >(11)where < τ1|τ2 > = min(t1, t2) .We now use (11) to construct a q-deformation of Brownian motion. Let (Ω, Ft, µ, Bt) bethe usual Brownian motion.
Ωis the set of continuous functions vanishing at t= 0, µ isthe Wiener measure and Ft is the σ-ring generated by { Bs : 0≤s ≤t}. On the other handlet (H, At, ψ0, φ1(t)) be the free quantum field over K= L2([0, ∞), R).
φ1(t) = φ1(χ[0,t])(for q= 1), H is the symmetric Fock space over K, ψ0 is the Fock vacuum and At isthe W ∗−algebra generated by { φ1(s) : 0 ≤s ≤t}. Then[16], interpreting Bt as amultiplication operator in L2(Ω, Ft, µ), there is a unitary operator V : L2(Ω, Ft, µ) →Hsuch that V BtV −1 = φ(t).
I.e. φ(t) as a stochastic process with expectationE(f(φ(t))) =< ψ0, f(φ(t))ψ0 >(12)coincides with Brownian motion.
For this identification of the free scalar field with Brow-nian motion it is useful to characterize the filtration At by the conditional expectation ofWick products[21]E(: φ1(u1)...φ1(un) : |At) =: φ1(χ[0,t]u1)...φ1(χ[0,t]un) :(13)2
Recall that the Wick products span the algebra generated by φ1(u). Hence, by linearity,definition on Wick products suffices to define conditional expectations on the completealgebra.We now use a minimal version of this correspondence to define q -deformed Brownianmotion.Definition: q-deformed Brownian motion is the process (Ω, Ft, µψ0, φq(t)) where(i) φq(t) is the operator defined by (9−11)(ii) Expectations of field functionals f(φq) are obtained byE(f(φq)) =< ψ0, f(φq)ψ0 >(14)ψ0 being defined by aqψ0 = 0(iii) The filtration Ft is characterized by the conditional expectations of Wick productsE(: φq(t1)...φq(tn) : |Fs) =: φq(χ[0,s]χ[0,t1])...φq(χ[0,s]χ[0,tn])(15)−−−−−−−−−−−−−−Notice that the algebraic relations (11) allow all elements of the algebra generated by φq(t)to be reduced to Wick products, hence all conditional expectations may be computed.Notice also that in this minimal definition the family Ft of measurable events is fixed inadvance and we avoid an explicit realization of the probability space Ω.Theorem 1 : q-deformed Brownian motion has :(i) zero mean, E(φq(t)) = 0(ii) variance E(φq(t) φq(s)) =min(s, t)(iii) independent increments in the senseE({φq(t1) −φq(t2)}{φq(t3) −φq(t4)}) = E(φq(t1) −φq(t2))E(φq(t3) −φq(t4)) = 0if there is no overlap between the intervals [t1, t2] and [t3, t4](iv) is a martingaleProperties (i) to (iii) follow by a simple computation using (10) (11) and (14).Themartingale property is a consequence of (15).
If s < tE(φq(t)|Fs) = φq(χ[0,s]χ[0,t]) = φq(s)Theorem 1 summarizes the similarities of q-deformed Brownian motion to the usual Brow-nian motion. The next result displays their main differences as well as an explicit char-acterization in terms of cumulants.3
Theorem 2 :(i) q-deformed Brownian motion is not a Gaussian process(ii) The cumulants areET(φq(t1)...φq(tn)) =Xc(i1i2)...(in−1in) < ti−1|ti2 > ... < tin−1|tin >(16)where < ti|tj >=min(ti, tj) , the sum is over all different partitions of the set (t1...tn) intopairs and the coefficients c(i1i2)...(in−1in) are obtained by the following graphical rules(ii.a) For each term in (16) one draws the < tik| tik+1 > contractions as follows❞❞❞❞❞❞❞❞❞❞...(17)XX2112345678910(ii.b) For each crossing of lines there is a factor { 12(q + q−1) −1} in c(i1i2)...(in−1in)(ii.c) For each contraction contained at depth α inside other contractions there is a factor{ 12(qα + q−α) −1} in c(i1i2)...(in−1in)(ii.d) If there are no crossings nor inner contractions the coefficient c(i1i2)...(in−1in) vanishes.−−−−−−−−In the example (17) the contribution of the diagram to the coefficient is{12(q + q−1) −1}2{12(q2 + q−2) −1}{12(q + q−1) −1}the first factor coming from the crossings and the last two from the inner contractionsat level 2 and 1. A necessary and sufficient condition for a process to be Gausssian isthat it possesses cumulants of all orders and that they vanish for order higher than 2.Computing the 4-time correlation one obtains using (11) and (14)E(t1t2t3t4) = E(φq(t1)φq(t2)φq(t3)φq(t4)) =< t1|t2 >< t3|t4 > +12(q + q−1) < t1|t3 >< t2|t4 > +12(q + q−1) < t1|t4 >< t2|t3 >implying that the cumulantET(t1t2t3t4) = E(t1t2t3t4) −E(t1t2)E(t3t4) −E(t1t3)E(t2t4) −E(t1t4)E(t2t3)does not vanish.
Hence the process is not Gaussian.The explicit expression for the cumulants of arbitrary orders is obtained by systematic4
reduction of the expectation values using the algebraic relations (11). The “crossing lines”factor comes from the coefficient of the second term in the l.h.s.
of (11) and the “innercontractions” factor from the r.h.s. together with (8).As a final remark we point out that using q-fermions b and b† withbb† + qb†b = qM[M, b†] = b†[M, b] = −b(18)and a generalization along the lines of (6 -11) one may construct a q-deformation of thenon-commutative Clifford process[21].
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