q-Legendre Transformation: Partition Functions and Quantization of the Boltzmann Constant

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by  Artur E. Ruuge

q-LEGENDRE TRANSFORMATION: PARTITION
FUNCTIONS AND QUANTIZATION
OF THE BOLTZMANN CONSTANT
ARTUR E. RUUGE AND FREDDY VAN OYSTAEYEN
0
1
0 Abstract. In this paper we construct a q-analogue of the Legendre transforma-
2 tion,whereq isamatrixofformalvariablesdefiningthephasespacebraidingsbe-
tween the coordinates and momenta (the extensive and intensive thermodynamic
n
a observables). Ourapproachisbasedonananalogybetweenthesemiclassicalwave
J functions inquantum mechanicsandthe quasithermodynamicpartitionfunctions

4 instatisticalphysics. Thebasicideaistogofromtheq-Hamilton-Jacobiequation
1 in mechanics to the q-Legendre transformation in thermodynamics. It is shown,

thatthisrequiresanon-commutativeanalogueofthePlanck-Boltzmannconstants
]
A (~ and kB) to be introduced back into the classical formulae. Being applied to
Q statistical physics, this naturally leads to an idea to go further and to replace
the Boltzmann constant with an infinite collection of generators of the so-called
.
h epoch´e (bracketing) algebra. The latter is an infinite dimensional noncommuta-
t
a tive algebra recently introduced in our previous work, which can be perceived as
m an infinite sequence of “deformations of deformations” of the Weyl algebra. The
[ generatorsmentionedarenaturallyindexedby planarbinaryleaf-labelledtreesin

such a way, that the trees with a single leaf correspond to the observables of the
2
limiting thermodynamic system.
v
2
0
3
5
2. I. Introduction
1
9 The Legendre transformation plays a rather fundamental role in mathematical
0 physics. One can immediately think of two examples: phenomenological thermody-
:
v namics and classical mechanics. The free energy F and the internal energy E of a
i
X thermodynamic system are related to each other in the same way as the Lagrangian
r L and the Hamiltonian H of a mechanical system. At this level, the Legendre trans-
a
formation is just a convenient concept that connects different pictures of description
of a physical system.
Much more important is that the Legendre transformation is associated to the
transition from quantum statistical physics to the classical limit. It emerges both
in the semiclassical approximation of quantum mechanics, as well as in the qua-
sithermodynamic approximation of statistical physics. In the first case, one needs
to consider the (additive) asympotics of the wave functions of the coordinate and
momentum representations corresponding to ~ 0, and in the second case one
→
needs to consider the (multiplicative) asymptotics of the partition functions of the
canonical and microcanonical ensembles corresponding to k 0. There is an anal-
B
→
ogy between these two limits corresponding to a similarity between fast oscillating
wave functions and rapidly decaying partition functions.
2000 Mathematics Subject Classification. 81Q20,81S10, 82B30.
1
2 A.E. RUUGEAND F. VAN OYSTAEYEN
Theoriginalmotivationofthispaperwastostudyaq-deformationoftheLegendre
transformation(whereq isamatrixofformalvariables)fromtheperspective ofthese
limit transitions. It turns out, that the consequences of these investigations seem
to go much deeper than one would expect and indicate a necessity to replace the
fundamental constants ~ and k with infinite collections of quantities ~ and
B Γ Γ
{ }
(k ) , respectively. These collections correspond to the generators of the epoch´e
B Γ Γ
{ }
algebra introduced in [1], and are structured in a slightly more complicated way
than infinite dimensional matrices (the Heisenberg’s quantization). The index Γ
varies over a set of finite leaf-labelled planar binary trees of different sizes, while the
labelling set is just the set of symbols corresponding to the degrees of freedom of
the limiting physical system. We start with describing the four examples mentioned
in a little more detail.
Example 1. If F = F(T,V) is the free energy of a 2-dimensional thermodynamic
system at absolute temperature T and macroscopic volume V, then the correspond-
ing entropy S is given by S = (∂F/∂T) , and the corresponding pressure is given
V
−
by p = (∂F/∂V) . Taking the Legendre transformationof F(T,V) with respect to
T
−
T yields the internal energy E of the system in terms of S and V, E = E(S,V), and
the Legendre transformation with respect to V defines the Gibbs potential G(T,p).
The restriction of the 1-form
α := TdS pdV
−
tothesubmanifoldoftheequilibrium states Λ R4(S,V;T,p)isexact: α = dE ,
Λ Λ
⊂ | |
since (TdS) = δQ (the infinitesimal amount of heat absorbed by the system) and
Λ
|
(pdV) = δA (the infinitesimal work produced by the system) add up according to
Λ
|
the law of the conservation of energy to dE = δQ δA.
Λ
| −
Example 2. This basic example of the Legendre transformation stems from clas-
sical mechanics. If L = L(x,v) is the Lagrangian of a system with d degrees of
freedom described by coordinates x Rd and the associated velocities v Rd,
∈ ∈
then one may consider the momenta p = ∂L(x,v)/∂v , i = 1,2,...,d. Assuming
i i
that this implicitly defines v = v(x,p), p = (p ,p ,...,p ), one can switch to the
1 2 d
Hamiltonian formalism via the Legendre transformation
e
H(x,p) = (pv L(x,v)) ,
v=v(x,p)
− | e
where pv := d p v . Introducing f := ∂L(x,v)/∂x , i = 1,2,...,d, one may
i=1 i i i i
interpret the Pstates of the system as a 2d-dimensional (Lagrangian) submanifold
Λ# R4d(p,x;v,f), f = (f ,f ,...,f ), such that the restriction of the 1-form
1 2 d
⊂
d
ε := (v dp f dx )
i i i i
−
Xi=1
to Λ# is exact, the conservation of energy being nothing else but ε = dH .
Λ# Λ#
| |
The Legendre transformation becomes conceptually more important if one takes
a step from classical mechanics to quantum mechanics and from the phenomeno-
logical thermodynamics to the statistical physics of equilibrium states. It is quite
remarkable that these two steps are associated with a pair of fundamental physical
q-LEGENDRE TRANSFORMATION 3
constants
~ = 1.05 10−27erg s,
×
k = 1.38 10−16erg K−1,
B
×
the Planck’s constant ~ and the Boltzmann constant k . Naively, ~ corresponds
B
to the quantization of the spectra of “physical properties”, and k corresponds to
B
the quantization of the “chemical substance” itself. Let us consider the transition
from quantum mechanics to classical mechanics. This is equivalent to pretending
that ~ 0 (in fact, this will be a dimensionless combination of ~ and some scaling
→
parameters defining the natural units of measurement in the experimental set-up).
To avoid a confusion in what follows, one may wish to denote the physical values of
~ and k mentioned above as ~phys and kphys.
B B
Example 3. Suppose the Schro¨dinger equation for a given physical system is of
the shape
1
∂ψt(x) ∂
i~ ~ = H x2, i~ ψt(x),
~
∂t − ∂x
(cid:16) (cid:17)
where x Rd are the classical coordinates, t R is classical time, ψt(x) L2(Rd)
~
is the wa∈ve-function, H(x,p) is (for simplicity∈) a polynomial in x and p ∈Rd (the
∈
classical momenta), and the indices atop denote the order of action. If we switch
the representation of the algebra of observables to an isomorphic one by taking the
~-Fourier transform, ψt(p) = (2π~)−d/2 dx exp( ipx/~)ψt(x), then the shape of
~ ~
−
the Schro¨dinger equation is just as goodR:
e
2
∂ψt(p) ∂
i~ ~ = H i~ ,p1 ψt(p).
~
∂t ∂p
e (cid:16) (cid:17)
e
Suppose ψt(x) is described for some t [0,T] by an additive asymptotics ψt(x) =
~ ~
∈
ϕt(x)exp(iSt(x)/~)/ Jt(x) + O(~), where ~ 0, St(x) is a real-valued smooth
→
function in x and t,pϕt C∞(Rd), for each t [0,T], and Jt(x) = detSt (x) =
∈ 0 ∈ | xx | 6
0, for t [0,T] and x suppϕt, where St (x) := ∂2St(x)/∂x ∂x d is the
∈ ∈ xx k i jki,j=1
Hess matrix of St(x). Then ψt(p) = ϕt(p)exp(iSt(p)/~)/ Jt(p) + O(~), where
~
q
the functions St(p), Jt(p), and ϕt(p) are smooth and can be computed using the
e e e e
stationary phase method,
e e
e
1 ipx iSt(x) ϕt(x) iSt(p) ϕt(p)
dx exp + = exp +O(~),
(2π~)d/2 Z (cid:16)− ~ ~ (cid:17) Jt(x) (cid:16) e~ (cid:17) eJt(p)
p q
where O(~) are the terms of the formal asymptotic expansion in e~ 0 of order
→
> 1, Jt(p) = detSt , St := ∂2St(p)/∂p ∂p d , the radical sign denotes the
| pp| pp k i jki,j=1
principal square root. The function St(p) is just the Legendre transformation of
e e e e
St(x),
e
St(p) = ( px+St(x)) ,
x=xt(p)
− | e
where x = xt(p) is the soluetion of p = ∂St(x) with respect to x = (x ,x ,...,x )
∂x 1 2 d
if p = (p ,p ,...,p ) is perceived as a parameter. To link the functions ϕt(x) and
1 2 d
ϕt(p), itiscoenvenient toconsider thegraphΛt := (x,p) p = ∂St(x)/∂x Rd Rd,
{ | } ⊂ x× p
e
4 A.E. RUUGEAND F. VAN OYSTAEYEN
or, what is the same, Λt = (x,p) x = ∂St(p)/∂p Rd Rd, and then lift the
{ | − } ⊂ x × p
both functions to Λt with respect to the canonical projections π : Rd Rd Rd
e x x × p → x
and π : Rd Rd Rd, respectively. Denoting the result corresponding to ϕt as
p x × p → p
χt C∞(Λt), and the result corresponding to ϕt as χt C∞(Λt), one can check
∈ 0 ∈
using the explicit formulae of the stationary phase method, that χt and χt coincide
on Ut := (π )−1(suppϕt) up to a phase factor.eMoreeprecisely,
x Λt
| e
iπ
χt = exp M χt ,
Ut Ut
| 4 |
(cid:16) (cid:17)
where M Z. Computing this integer is actuallyequite important, since it leads to
∈
the discovery of theMaslov index. Upto this point, we have the thirdexample of the
Legendre transformation stemming from the semiclassical limit ~ 0 of quantum
→
mechanics. Informally, the Legendre transformation shows up as a fast oscillating
limit oftheFourier transform. It isinteresting tomention that it canbeperceived as
an idempotent analogue of the Fourier transform in the framework of “idempotent”
functional analysis and “tropical” algebraic geometry [2] if one replaces the usual
integrals with the idempotent integrals .
⊕
R R
Example 4. Finally, let us consider the fourth example of the Legendre trans-
formation. In analogy with the semiclassical approximation of quantum mechanics
where onedeals with ~ 0, thetransitionfromthestatistical physics ofequilibrium
→
states can be formally understood as a limit k 0. Indeed, consider a system
B
→
placed in a thermostat. Its equilibrium state is described by the partition function
1
Z (β) := exp βE ,
E∗ − k m
Xm (cid:16) B (cid:17)
where m is the index of a microscopic state, E is the corresponding energy,
m
E := (E ) , the sum is taken over all microscopic states, and β = 1/T is the
∗ m m
inverse temperature of the thermostat in the absolute thermodynamic scale. In the
thermodynamic limit (i.e. the macroscopic volume V , while the value of the
→ ∞
density of mass ρ > 0 is a fixed positive constant), for every fixed β = 1/T, we have
lnZ (β) = O(N), where N is the number of corpuscles in the system. If we
E∗ → ∞
choose a unit of measurement ε of the specific energy, then there is a small param-
0
eter k′ := k T/(Nε ) (see [3, 4, 5]). It can be expressed as k′ = k Tm /(Vρε ),
B 0 B 0 0
where m is the corpuscular mass. Since the parameters m , ε , T, and ρ are fixed
0 0 0
under the limit transition, essentially one deals with k /V. Therefore, k 0, V –
B B
→
fixed, is “the same thing” as k – fixed, V . The thermodynamic limit k′ 0
B
→ ∞ →
is formally equivalent k 0. Now, assuming the spectrum of energies is bounded
B
→
from below and satisfies min( E ) 0 as k 0, we have:
m m B
{ } → →
∞ 1
Z (β) = exp βE W (E,V)dE 1+o(k ) ,
E∗ Z − k kB · B
0 (cid:16) B (cid:17) (cid:8) (cid:9)
for some density W (E,V) of the measure of integration over dE, the param-
kB
eter V is the macroscopic volume of the system. In the equilibrium statistical
physics, the quantity F (T) := k β−1lnZ (β), β = 1/T, (termed the free
E∗ − B E∗
energy of the canonical Gibbs ensemble at temperature T = β−1) has a non-
vanishing limit as k 0, F (T) = F(T,V) 1+o(k ) . To ensure this, one takes
B → E∗ { B }
q-LEGENDRE TRANSFORMATION 5
W (E,V) = (A(E,V)/√2πk )exp(S(E,V)/k ) 1+o(k ) , forsome A(E,V)and
kB B B { B }
S(E,V). Then, the saddlepoint method yields:
1 ∞ 1 1
dE exp βE + S(E,V) A(E,V) =
√2πk Z − k k
B 0 (cid:16) B B (cid:17)
A(E,V) 1
= exp Ψ(β,V) 1+O(k ) , (1)
B
∂2S(E,V)/∂E2(cid:12)(cid:12)E=E(β,V) (cid:16)− kB (cid:17)·{ }
− e
p (cid:12)
where E = E(β,V) is the solutio(cid:12)n of equation β = ∂S(E,V)/∂E with respect to
E (perceiving β and V as parameters), and the function Ψ(β,V) is the Legendre
e
transformation of S(E,V) in the variable E,
Ψ(β,V) = (βE S(E,V)) .
− |E=Ee(β,V)
Since the leading term in the right-hand side of the previous equation (1) must
be just exp( βF(β−1,V)/k ), one concludes, that Ψ(β,V) = βF(β−1,V), and
B
−
A(E,V) = ∂2S(E,V)/∂E2. One has the fourth example of the Legendre trans-
−
formation.
Recall, that the transition k 0 from the equilibrium statistical physics to
B
→
the phenomenological thermodynamics is understood by identifying S(E,V) with
the entropy of the limiting thermodynamic system at the equilibrium state corre-
sponding to the internal energy E and the macroscopic volume V. Since on the
(Lagrangian) submanifold Λ R4(S,V;T,p) of the equilibrium states we have
⊂
dS = ((1/T)dE + (p/T)dV) , where p is the macroscopic pressure, and T is
Λ Λ
| |
the absolute temperature, an elementary computation yields:
∂2S(E,V) ∂ 1 1 ∂T 1 1
= = = = ,
− ∂E2 −(cid:16)∂E T(cid:17)V T2(cid:16)∂E(cid:17)V T2 ∂E T2cV(T,V)
∂T
(cid:16) (cid:17)V
where c (T,V) is the isohoric heat capacity. The assumption that we actually
V
need to be able to apply the saddlepoint method ∂2S(E,V)/∂E2 > 0 reduces
−
to c (T,V) > 0, which is one of the two conditions (along with (∂p/∂V) < 0)
V T
of stability of the equilibrium state (T,V) with respect to quasithermodynamic
fluctuations.
Now let us turn to the possibility of defining a q-generalization of the Legendre
transformation having the context of the four examples described above. Let q =
q 2s be a 2s 2s matrix of formal variables q satisfying
k i,jki,j=1 × i,j
q = 1, q q = 1, (2)
i,i i,j j,i
where i,j = 1,2,...,2s. Consider (over the basefield K) an algebra defined by
q
A
2s generators ξ ,ξ ,...,ξ and relations
1 2 2s
ξ ξ = q ξ ξ , (3)
i j j,i j i
where i,j = 1,2,...,2s (the quantum affine space). This algebra is Z-graded,
= n, where n is formed by degree n homogeneous polynomials, if n > 0,
Aq n∈ZAq Aq
and wLe put n in case n < 0. Let be the completion of with respect to the
Aq Aq Aq
canonical increasing filtration F• associated to this grading, FN := n.
Aq b Aq n6N Aq
If we think about the first s generators ξ1,ξ2,...,ξs as momenta (andLredenote
them as ξ = p ,ξ = p ,...,ξ = p ), and the other s generators ξ ,ξ ,...,ξ
1 1 2 2 s s s+1 s+2 2s
6 A.E. RUUGEAND F. VAN OYSTAEYEN
as coordinates (and redenote them as ξ = x ,ξ = x ,...,ξ = x ), can we
s+1 1 s+2 2 2s s
define an analogue of the Legendre transformation? The first two examples above
imply, that one could think of this analogue as a map : . On the other
q q q
L A → A
hand, the third and the fourth examples are slightly of different nature: there is an
extra parameter introduced in the story (the Planck constabnt ~ inb example three,
and the Boltzmann constant k in example four). This suggests, that the required
B
analogue could be a map LBq : Aq ⊗K B → Aq ⊗K B, where B is another algebra
extending the scalars, or even more general, a map : A , where is a
b b q q q q
L → A A
filtered algebra such that the degree zero of its associated graded is isomorphic to
e e e e
. The answer suggested in the present paper corresponds (more or less) to this
q
A
third possibility.
In one of our recent papers [1], among other things, we were interested in motivat-
ing a q-analogue of the Weyl quantization map W in quantum mechanics. Recall,
that the map W can be described as follows. One considers two algebras, a commu-
tative algebra of polynomials in x ,x ,...,x and p ,p ,...,p , and the Heisen-
1 2 s 1 2 s
A
berg algebra generated by x ,x ,...,x ,p ,p ,...,p , and relations [p ,x ] = h,
1 2 s 1 2 s j k
A
[p ,p ] = 0, [x ,x ] = 0, and [x ,h] = 0, [p ,h] = 0, where [ , ] denotes the
j k j k j k
commutator, abnd j,k = 1,2,b...b,s. Inbtheb“cboordinabte represen−ta−tionb” xb x
j j
(bmubltiplicationb), pb i~∂/∂x , j,k = 1,2,...,s, and the central generator h7→cor-
k k
responds to the mult7→ipl−ication by i~. The map W is just a linear map W b:
b − A → A
implementing the symmetrization over the “order of action” of the quantized coor-
b
dinates and momenta,
1
W(z z ...z ) := z z ...z ,
i1 i2 in n! iσ(1) iσ(2) iσ(n)
X
σ∈Sn
b b b
whereS isthesymmetricgrouponnsymbols(nisapositiveinteger), i ,i ,...,i
n 1 2 n
∈
1,2,...,2s , and z := p , z := x , and z := p , z := x , for j = 1,2,...,s.
j j s+j j j j s+j j
{ }
Now, being interested in a q-analogue of W : , it was natural to introduce
b Ab→bA b
theso-called“non-commutativePlanckconstants”. Note,thatasimilarconstruction
b
motivated by the superstring theory appears in [6]. The q-analogue mentioned
is understood as a vector space map W : between the two algebras.
q q q
A → A
The first one is the affine quantum space , denote the generators ξ ,ξ ,...,ξ ,
q 1 2 2s
A b
ξ ξ = q ξ ξ , where q are formal variables satisfying the usual assumptions (2),
i j j,i j i i,j
i,j = 1,2,...,2s. The second algebra has generators ξ , i = 1,2,...,2s, and h ,
q i j,i
A
1 6 i < j 6 2s. We extend the notation h for any i,j = 1,2,...,2s, by h := 0
b j,i b i,i b
and h := q−1h . Part of the relations is the deformation of the relations (3) for
i,j − j,i j,i b b
the algebra ,
A b
ξ ξ q ξ ξ = h ,
i j j,i j i i,j
−
for any i,j = 1,2,...,2s. Let us order the generators as follows: ξ ξ , if i < j,
bb bb b i j
≺
ξ h , for any i and any i′ < j′, and h h , if i < i′, or if i = i′ and j < j′
i ≺ j′,i′ j,i ≺ j′,i′ b b
(where i,j,i′j′ 1,2,...,2s ). This algebra has a Poincar´e-Birkhoff-Witt basis
b b ∈ { } b b
(with respect to ) if we impose the braidings
≺
ξ h = q q h ξ ,
i j′,i′ j′,i i′,i j′,i′ i
hj,ihj′b,i′b= qj′,jqj′,iqi′,jbqi′,ihbj′,i′hj,i,
b b b b
q-LEGENDRE TRANSFORMATION 7
for any i,j,i′,j′ 1,2,...,2s . With these relations, we obtain an algebra for
q
∈ { } A
which there exists a reasonable analogue W : of the Weyl quantization
q Aq → Aq b
map (the generators ξ ,ξ ,...,ξ are the “q-quantized” coordinates and momenta.)
1 2 2s b
Note, that the algebra is naturally filtered, with the filtration F• defined by
b Abq b Aq
ξ F0 , h F1 F0 .
i ∈ Aq j,i ∈ Aq\b Aq b
It turns out that this idea to introduce the additional generators with non-trivial
b b b b b
braidings is quite useful in the construction of the q-Legendre transformation. In
other words, the generalization we suggest corresponds to the examples three and
four described above which involve the fast oscillating integrals (~ 0, the station-
→
ary point method), or rapidly decaying integrals (k 0, the saddle point method).
B
→
The plan of the attack is more or less as follows. We start with the Hamilton-Jacobi
equation
∂St(x)/∂t+H(x,∂St(x)/∂x) = 0,
which is a non-linear equation from classical mechanics (x Rs are the coordinates
of a system, t R is time, H(x,p) is the Hamiltonian, ∈p Rs are the canoni-
∈ ∈
cally conjugate momenta corresponding to x, St(x) is the action as a function of
(the ending point) coordinates and time). As is well known, the solution of the
Hamilton-Jacobi equation can be described in terms of a system of ordinary dif-
ferential equations x˙ = ∂H(x,p)/∂p, p˙ = ∂H(x,p)/∂x with the initial conditions
−
x = α, p = ∂S0(α)/∂α (the Hamiltonian system). We investigate how far
t=0 t=0
| |
can we go in generalizing this fact if the Hamilton-Jacobi equation is replaced with
its analogue constructed in a certain way over the “braided” generators ξ and
i i
{ }
h . If the Hamilton-Jacobi equation is perceived as a classical limit ~ 0
{ j,i}i<j b →
of the Schro¨dinger equation, then at this point one realizes that it is necessary to
b
introduce the “Planck constants” h back in the equation in order to “control the
j,i
braidings” between the symbols in the corresponding formulae. The next step is to
b
look at the fact that the Legendre transformation St(p) of St(x) satisfies again the
Hamilton-Jacobi equation,
e
∂St(p)/∂t+H( ∂St(p)/∂p,p) = 0.
−
Let Λt := (x,p) p = ∂Set(x)/∂x . Theneon Λt Rs Rs we have St(p) =
{ | } ⊂ x × p |Λt
( px+St(x)) . It turns out that there is some problem to generalize this fact to
− |Λt e
the q-deformed case. At that point we will have to make precise, what we mean by
the q-Legendre transformation, but whatever it is, it is important to point out, that
the construction still involves the additional generators having a purpose to control
thebraidingsintheformulae. Itremains tomake thethirdsmall step. Oncewe have
the q-Legendre formulae, there is no need to interpret the auxiliary generators h as
j,i
having a quantum mechanical origin. For example, if we specialize all the braidings
b
as q = 1 (and by that everything becomes commutative), then the analogue of the
i,j
Legendre transformation can be perceived as some construction involving the tensor
algebra T( 2(V)), where V is the vector space of linear functions on Rs Rs. There
x× p
is no moreVreason to view h as non-commutative Planck constants, but one could
i,j
say that they could be the non-commutative (q-commutative) Boltzmann constants.
b
Denote them (k ) . This suggests that if one is interested in q-deforming the
B i,j
statistical physics of equilibrium states (or, more generally, in its R-braiding, where
b
8 A.E. RUUGEAND F. VAN OYSTAEYEN
R is a solution of the Yang-Baxter equation), then it is necessary to introduce the
non-commutative Boltzmann constants (k ) in place of k .
B ∗ B
b
II. q-analogue of the Hamilton-Jacobi equation
Let us start with a description of the q-analogue of the phase space and the q-
analogueofthePoissonbracket. Fixapositiveintegersandconsidera2s 2smatrix
×
q of formal variables q satisfying q = 1 and q q = 1, for k,l 1,2,...,2s .
k,l k,k k,l l,k
∈ { }
Consider an algebra defined by generators p ,p ,...,p ;x ,x ,...,x , and rela-
q 1 2 s 1 2 s
A
tions
b b b b b b
p p = q p p , p x = q x p , x x = q x x ,
i j j,i j i i α s+α,i α i α β s+β,s+α β α
where i,j,α,β = 1,2,...,s, (the q-affine phase space). We will also use the notation
bb b b bb b b b b b b
ξ ,ξ ,...,ξ (without hats), and set ξ := p , if k 6 s, and ξ := x , if k > s
1 2 2s k k k k−s
(k 1,2,...,2s ). Extend this algebra by adding more generators h , i,α =
α,i
1,2∈,..{.,s, satisfyi}ng b b
p h = q q h p , x h = q q h x ,
i α,j s+α,i j,i α,j i α β,i s+β,s+α i,s+α β,i α
where i,j,α,β = 1,2,...,s. Denote the extended algebra and equip it with a
b b b q b
A
bracket
e
, : ,
q q q
h− −i A ×A → A
defined as follows. Let , be a bilinear map, such that
e e e
h− −i
p ,x := h , x ,p := q p ,x ,
i α α,i α i i,s+α i α
h i h i − h i
p ,p := 0, x ,x := 0,
b b h i ji b bh α βi b b
where i,j,α,β = 1,2,...,s. Extend it to the higher order monomials in p and x
b b b b i α
as a q-biderivation,
b b
m n
ξ ...ξ ,ξ ...ξ := Qj1,...,jn(µ,ν) ξ ,ξ ξ ...ξˇ ...ξ ξ ...ξˇ ...ξ ,
h im i1 jn j1i i1,...,im h µ νi im iµ i1 jn jν j1
Xµ=1Xν=1
where i ,...,i ,j ,...,j 1,2,...,2s , the check symbol atop ξ and ξ means
1 m 1 n ∈ { } iµ jν
that the corresponding factors are omitted, and the braiding factor Qj1,...,jn(µ,ν) is
i1,...,im
defined from
ξ ...ξ ξ ...ξ = Qj1,...,jn(µ,ν)ξ ξ ξ ...ξˇ ...ξ ξ ...ξˇ ...ξ .
im i1 jn j1 i1,...,im µ ν im iµ i1 jn jν j1
Finally, extend the bracket to the monomials containing h , α,i = 1,2,...,s, as
α,i
h f,g := h f,g , f,gh := f,g h ,
α,i α,i α,i α,i
h i h i h i h i
for any monomials f,g . This yields a bracket , on the algebra which
q q
∈ A h− −i A
can be regarded as a q-analogue of the Poisson bracket , “not divided by the
e {− −} e
Planck constant ~”. Define q , for any monomials f,g , by fg = q gf. It is
g,f q g,f
∈ A
straightforward to check that , satisfies the q-Jacobi identity,
h− −i e
F, G,H +q q G, H,F +q q H, F,G = 0,
G,F H,F H,F H,G
h h ii h h ii h h ii
for any monomials F,G,H in the generators ξ and h .
q i i α,j α,j
∈ A { } { }
Let us now recall, how the Hamiltonian system of equations emerges from the
e
Hamilton-Jacobi equation in the classical mechanics with one degree of freedom
q-LEGENDRE TRANSFORMATION 9
(s = 1). We have an unknown function S(x,t) (the action) in coordinates x R
and time t R. The Hamilton-Jacobi equation looks as follows: ∈
∈
∂S(x,t)/∂t+H(x,∂S(x,t)/∂x) = 0,
where H is a smooth function in coordinate x and momentum p, H : R R R,
x p
× →
(the Hamiltonian function), the momentum p corresponds to ∂S(x,t)/∂x. Suppose
we have a smooth R-valued function X(y,t) in a R-valued variable y and time t,
and evaluate ∂S(x,t)/∂x at x = X(y,t). Then for P(y,t) := (∂S(x,t)/∂x)
x=X(y,t)
|
we have:
∂P(y,t) ∂2S(x,t) ∂2S(x,t) ∂X(y,t)
= + =
∂t (cid:16) ∂x∂t (cid:17)(cid:12)x=X(y,t) ∂x2 (cid:12)x=X(y,t) ∂t
(cid:12) (cid:12)
∂ ∂(cid:12)S(x,t) (cid:12) ∂2S(x,t) ∂X(y,t)
= H x, + =
(cid:16)− ∂x (cid:16) ∂x (cid:17)(cid:17)(cid:12)x=X(y,t) ∂x2 (cid:12)x=X(y,t) ∂t
(cid:12) (cid:12)
∂H(x,p) (cid:12) ∂2S((cid:12)x,t)
= +
−(cid:16) ∂x (cid:12)p=∂S(x,t)/∂x(cid:17)x=X(y,t) ∂x2 (cid:12)x=X(y,t)×
(cid:12) (cid:12)
(cid:12) ∂H(x,p) (cid:12) ∂X(y,t)
+ .
×h−(cid:16)(cid:16) ∂p (cid:17)(cid:12)p=∂S(x,t)/∂x(cid:17)(cid:12)x=X(y,t) ∂t i
(cid:12) (cid:12)
To satisfy this, it suffices to put (cid:12) (cid:12)
∂P(y,t) ∂H(x,p) ∂X(y,t) ∂H(x,p)
= , = ,
∂t − ∂x (cid:12) ∂t ∂p (cid:12)
(cid:12)x=X(y,t),p=P(y,t) (cid:12)x=X(y,t),p=P(y,t)
(cid:12) (cid:12)
which is just the Hamilton(cid:12)ian system of equations. (cid:12)
Turnnowbacktothealgebra . Wehavethe“braided”coordinatesx ,x ,...,x ,
q 1 2 s
A
the “‘braided” momenta p ,p ,...,p , and the auxiliary generators h s (the
“q-distortion” of the Planc1k c2onestants ~). To generalize the Hamilto{n-bJαa,i}cboαb,i=i1equba-
b b b
tion to the algebra , one should generalize the symbolic computation linking the
q
A
classical Hamilton-Jacobi equation with the Hamiltonian system of equations defin-
e
ing the phase space trajectories. Observe, that in the classical case with s = 1, the
derivative ∂S(x,t)/∂x can be expressed in terms of the canonical Poisson bracket
, as ∂S(x,t)/∂x = p,S(x,t) . Introduce the Planck constant ~ back into the
{− −} { }
classical equations:
∂S(x,t) 1
= ~ p, S(x,t) ,
∂x ~
n o
1∂S(x,t) 1 1
+ H x,~ p, S(x,t) = 0.
~ ∂t ~ ~
(cid:16) n o(cid:17)
One can see, that everything can be expressed in terms of the rescaled functions
S(x,t) = S(x,t)/~, H(x,p) = H(x,p)/~, and the Poisson bracket not divided by ~,
f,g = ~ f,g (where f and g are any smooth functions in (x,p)). It is natural to
hb i { } b
think that the analogue of the Hamilton Jacobi equation for the algebra must
q
A
be of the shape
e
∂
St(x)+H x, p,St(x) = 0, (4)
∂t h i
(cid:0) (cid:1)
but what is St(x) and H(x,pb) inbthisbcabse?b b b
b b b b b
10 A.E. RUUGEAND F. VAN OYSTAEYEN
The basic idea can be formulated as follows. In the classical case (for simplicity,
s = 1), we have the functions S(x,t), H(x,p), and X(y,t), which can be expanded
into formal power series:
∞ tmxN ∞ xKpL ∞ tmyN
(m) (m)
S(x,t) = b , H(x,p) = T , X(y,t) = X ,
m!N! N K!L! K,L M!N! N
X X X
m,N=0 K,L=0 m,N=0
(m) (m)
where b , T , and X are some coefficients. Let us do everything in terms of
N K,L N
power series. In particular, the left-hand side of the Hamilton-Jacobi equation can
be formally expanded into such series, yielding an infinite collection of links between
the mentioned coefficients. In the multidimensional case (i.e. any s), the formulae
are totally similar, but one needs to use the standard notation with multi-indices
N = (N ,N ,...,N ), N! = N !N !...N !, xN = xN1xN2 ...xNs, etc. Informally,
1 2 s 1 2 s 1 2 s
the approach reduces to an imperative: expand everything into power series and
then make the coefficients non-commutative.
To illustrate the corresponding computation, assume for simplicity that q = 1
i,j
and q = 1, for i,j,α,β = 1,2,...,s (the other q can be non-trivial). Then,
s+α,s+β i′,j′
for example, p ,x x = h x +h x (for any i,α,β = 1,2,...,s), since one can
i α β α,i β β,i α
h i
freely permute x and x . One obtains:
α β
tmxN ∞ tm s N h xN−1µ
St(x) = b(m), p ,St(x) = µ µ,i b(m),
m!N! N h i i m! N! N
mX,N b mX=0 Xµ=1NX>0 b
b b b b b
where 1 := (0,...,0,1,0,...,0) with 1 standing in the µ-th position, the notation
µ
N > 0 is understood as N > 0 for every ν = 1,2,...,s. The time t is kept as a
ν
commutative variable, although it is possible to consider some braidings involving t
aswell. Now, since thereisafactorN inthenumerator, onecanassume that forN
µ µ
the summation starts with 1, but not 0. Changing summation index to N′ = N+1 ,
µ
and then leaving out the prime in the final expression, we obtain:
s ∞ tmxN
p ,St(x) = p ,x b(m) ,
h i i h i µi m!N! N+1µ
Xµ=1 mX=0NX>0 b
b b b b b
where we go back to p ,x = h Similarly,
i µ µ,i
h i
b b∂ St(x) = ∞ tmxNb(m+1).
∂t m!N! N
mX=0NX>0 b
b b
Wewouldlikethequantity p ,St(x) to“behavelike” p withrespect tothebraiding
i i
h i
coefficients. This implies that, for every µ, m and N, p ,x xNb(m) behaves like
b b bh i µi N+1µ
p . Since the “braiding behaviour” of p ,x is like of the product p x = p x1µ, one
i i µ i µ i
h i b
can conclude, that for any m and N, the coefficient b(m) behaves like x−N. Making
b N
it more formal, impose the following rules:
b
s s
p b(m) = q−Nα b(m)p , x b(m) = b(m)x , h b(m) = q−Nα b(m)h ,
i N s+α,i N i β N N β β,i N s+α,i N β,i
(cid:16)αY=1 (cid:17) (cid:16)αY=1 (cid:17)
b b b b (5)
for any i,β = 1,2,...,s, any m Z , and any N = (N ,N ,...,N ) > 0.
>0 1 2 s
∈