V. Fock1
temporarily in G¨ottingen
Received 1 May 1928
Zs. Phys.49, 339, 1928 Fock57, pp. 9–24
In his paper “Emission and Absorption of Radiation” [1] Dirac considers the Bose–Einstein statistics of an ensemble of mechanical systems by a completely new method. He admits that the perturbation of the system by an external force takes place and considers the changes caused by this perturbation in the given probability distribution of the energy levels in the ensemble.
What is essentially new in the Dirac method is that he considers the number Ns of systems on thes-th energy level as a canonical variable.
In the space of such variables (which we’ll call the Dirac space) Dirac establishes the wave equation; its solutions are the functions ofNsand time. The square modulus of the solution defines the probability of the corresponding distribution of the energy levels in the ensemble.
However an algorithm on how to solve this equation is not given in the Dirac paper.
In the present paper2the problem is generalized and we look for the probability distribution of any arbitrary mechanical quantity (not only of energy), whereas the probability amplitudes for the distribution of another (or the same) arbitrary mechanical quantity at time t= 0 are given as initial conditions.
1International Education Board Fellow.
2In the original text an attempt has been made to apply the method developed in this paper to the Fermi statistics. This attempt was unsuccessful, and therefore the parts related to the Fermi statistics (including§8) are omitted. The erratum was published in the paper “On Quantum Electrodynamics” Sow. Phys. 6, 5, 428 (1934). The difficulties, connected with the application of this method to the Fermi statistics, were mentioned already in the original paper (V. Fock, 1957). The present translation uses the revised version in Fock57. (Translator)
The generalized Dirac equation is solved generally by the use of the generating function for which an explicit expression can be given.
§1. We consider an ensemble of identical mechanical systems. The energy operator H of each system can contain time. Together with energy we also consider two other mechanical quantities a and b with operatorsAandB.
Further we need operatorA only for a fixed time t0 = 0, so we can assume thatAdoes not depend on time explicitly. On the contrary, we consider operatorB at a variable time and correspondingly we assume that it can explicitly contain time.3
The Schr¨odinger equation for a single system is Hψ+~
i
∂ψ
∂t = 0. (1)
The eigenfunctions of operatorsAandB satisfy the equations A ψs(q) =asψs(q), (2) B ϕs(q, t) =βs ϕs(q, t). (3) The arbitrary time-dependent phase factors of functionsϕs(q, t) can be chosen by the prescription4proposed by the author in [2].
We will also consider a system of solutionsψs(q, t) of the Schr¨odinger equation
Hψs(q, t) +~ i
∂ψs
∂t = 0 (4)
that satisfy the initial conditions
ψs(q,0) =ψs(q). (5) According to the theorem proved in the quoted paper by the author the system of functionsψs(q, t) will be complete, normalized and orthogonal for anyt. Further we will call it the basic system.
§2. Each solution f(q, t) of the Schr¨odinger equation (1) is defined uniquely by its initial valuef(q,0). If we expand the initial value into the set of functionsψs(q) =ψs(q,0):
f(q,0) =X
s
gs ψs(q,0), (6)
3In Dirac’s paper both operatorsAandBcoincide and are equal to the energy operator of the unperturbed system. (V. Fock)
4Each eigenfunction normalized along this prescription should be orthogonal to its time derivative. (V. Fock)
then at timetthe solution will be f(q, t) =X
s
gs ψs(q, t). (7)
The same solution can be expanded into eigenfunctions of operatorsA andB:
f(q, t) =X
s
xs ψs(q,0), (8)
f(q, t) =X
s
ysϕs(q, t), (9)
where coefficients xs and ys are functions of time, whereas gs in (7) were constants.5 The infinite-dimensional space of all sets of expansions’
coefficients
g1, g2, . . . gs, . . . x1, x2, . . . xs, . . . y1, y2, . . . ys, . . .
(10)
is called the complex Hilbert space. The elements of each line in (10) will be the “coordinates” in this space. Transition from one set (line) to another corresponds to a linear orthogonal (unitary) transformation of coordinates, which can be considered as a rotation of a coordinate system in the Hilbert space.
The physical meaning ofysis the following. Let the operatorbwith eigenvalues βs and eigenfunctions ϕs(q, t) correspond to the physical quantity b. If we expand the solution of the Sch¨odinger equation into functionsϕs(q, t), the square modulus|ys|2 of the expansion coefficient ysgives a relative probability for the quantity bto have the valueβsat timet.
Instead of saying “the quantity b is equal to the eigenvalue βs of operator B” we can say shorter “the system is in state s” each time when it is clear by which operator the states we are talking about are defined.
Because the sum of the squared moduli |ys|2 is equal to the corre- sponding sum of gs and therefore is a constant, we can put it equal to the number of systemsN in the whole ensemble
X
s
|ys|2=X
s
|gs|2=N. (11)
5If we considerψs(q,0) as eigenfunctions of an unperturbed system, thenxsare the same values that Dirac callsbs. (V. Fock)
The square modulus
|ys|2=Nsβ (12) is then the probable number Nsβ of systems in state s (i.e., with the eigenvalueβs). The same is naturally true also for operatorA.
§3. Let us establish the differential equations which are satisfied by the expansion coefficientsys.
If we multiply each of expansions (7) and (9) by ϕr(q, t)%(q)dq
[%(q) is the density function inq-space], integrate over q-space and put both results equal, then we obtain the expression for yr through con- stantsgs:
yr=X
s
Yrs gs, (13)
where for brevity we denoted Yrs =
Z
ϕr(q, t)ψs(q, t)%dq . (14) The valuesYrssatisfy the equations
X
l
YrlYsl=δrs
X
l
YlrYls=δrs
(15)
and therefore are the entries of the unitary matrix.
Differentiating (14) by time and taking into account that ψs(q, t) satisfies the Schr¨odinger equation (4), we obtain
dYrs
dt = Z ∂ϕr
∂t ψs%dq− i
~ Z
ϕrHψs %dq . (16) Expanding hereψs(q, t) intoϕl(q, t):
ψs(q, t) =X
l
Yls ϕl(q, t), (17)
and denoting for brevity Krl=
Z
ϕr(q, t)Hϕl(q, t)%dq+~ i
Z
ϕr(q, t)∂ϕl
∂t %dq , (18)
we get the differential equations
~ i
dYrs
dt =−X
l
KrlYls. (19)
The coefficients matrix Krl is evidently Hermitian. Because ys are the linear functions ofYrs with constant coefficientsgs, the ys satisfy the same differential equations (19). This statement can be presented as a theorem:
Theorem 1. The expansion coefficients ys of the solution of the Schr¨odinger equation into an orthogonal system ofϕs(q, t) are solutions of the differential equations
~ i
dyr
dt =−X
l
Krl yl, (20)
with the definition (18) ofKrl.
This theorem is evidently valid for any complete orthogonal system.
Particularly for the basic system (e.g., forψr(q, t)) expressions (18) are equal to zero and the expansion coefficients are constants.
§4. Together with the system of differential equations (20) we con- sider its complex conjugate one
~ i
dyr dt =X
l
Klr yl. (20∗)
Denoting the bilinear form byF, F =X
sr
Ksr ysyr, (21)
we can write equations (20) and (20∗) as
~ i
dyr
dt =−∂F
∂yr (22)
or ~
i dyr
dt = ∂F
∂yr. (22∗)
If we consideryr (or yr) as a canonical coordinate and ~
iyr (or −~ iyr) as a canonical momentum, i.e., if we puty
Qr=yr, Pr= ~
i yr (23)
or
Qr=yr, Pr=−~
i yr, (23∗)
equations (22) and (22∗) can be considered as canonical equations of motion in the Hilbert space with the Hamilton functionF.
Now following Dirac we will consider that canonical variables are operators (matrices,q-numbers) and establish the Schr¨odinger equation corresponding to the Hamilton operatorF. We can use here either the space ofyr or the space ofyr.
In the space ofyrthe operatoryrmeans “multiplication byyr” and the operatoryr means “changing the sign and taking the derivative by yr”:
yr→yr; yr→ − ∂
∂yr. (24)
In the space ofyrthe operatoryrmeans “multiplication byyr” and operatorys means “taking derivative byyr”:
yr→yr ; yr→ ∂
∂yr. (25)
This follows from the well-known general formulae Pr= ~
i
∂
∂Qr ; Qr=−~ i
∂
∂Pr, (26)
and is equally valid whether we define the canonical variables explicitly according to (23) or (23∗).
Denoting the wave function in space ofyr by Ω we get
~ i
∂Ω
∂t −X
sr
Ksr yr∂Ω
∂ys = 0. (27)
We have to remark here that the sequence of operators in the diagonal terms of (27) is not essential. Indeed, if we apply the operatorsyr and
∂
∂yr in a different sequence, and, e.g., write
∂
∂yr
(yrΩ) or 1 2
ã ∂
∂yr
(yrΩ) +yr∂Ω
∂yr
á ,
then instead of zero in the right-hand side of (27) we would have a term of the form c(t)Ω, where c(t) is a real function of time. However it is easily seen that the solution of the new equation differs from that of
(27) only by a factor with the absolute value equal to unity, i.e., by an inessential phase factor.
Equation (27) represents a wave equation in space ofyr. To obtain the corresponding equation in the Hilbert space withyras coordinates we have to substitute expressions (24) for operatorsyr, yr in the Hamilton function (21) by the expressions (25). If we denote the wave function in the spaceyr as Ω, we find
~ i
∂Ω
∂t +X
sr
Ksr ys ∂Ω
∂yr = 0. (28)
Due toKsr=Krs this equation is exactly complex conjugate to (27).
Thus, to get the wave equation in the Hilbert space it is unimportant whether we consider the expressions (23) or (23∗) as “coordinates” and
“momenta.” As the sequence of operators in the Hamilton function is also unimportant, we can state that the wave equation in Hilbert space is established uniquely.
Now we return to equation (27). This is a linear partial differential equation of the first order. If we form the system of ordinary differential equations corresponding to (27) we will exactly get the system (20). So this system and equation (27) are adjoint in the sense of the theory of partial differential equations. From this it follows that to make the transformation of wave equation (25) to a different coordinate system in the Hilbert space we need only to change the independent variables according to the ordinary rules of differential calculus. The wave function Ω is covariant relative to such transformation.
This remark allows one to find the general solution of wave equation (27) without any calculations. Indeed, if we choose the expansion coef- ficients ofgsinto the basic system of functions as the coordinates in the Hilbert space, then in these coordinates the wave equation has a simple
form à
∂Ω
∂t
ả
gs
= 0, (29)
where the subscriptgsmeans that the time derivative is taken at constant gs. The general solution of this equation is an arbitrary function of variablesgs:
Ω = Ω(g1, g2, . . . gs . . .), (30) where in the case of equation (25) we have to expressgs throughyr:
gs=X
r
Yrs yr. (31)
The general solution of equation (28) is complex conjugate to (30). We can summarize the results of this section as a theorem.
Theorem 2.The Dirac statistical equation in the Hilbert space with the coordinatesyr is a first-order partial differential equation adjoint to the system of ordinary differential equations (20) of Theorem 1. Its general solution will be an arbitrary function of the expansion coefficients into the basic system of functions.
§5. As we mentioned in§2, notysthemselves but their square moduli
|ys|2 =ysys (namely, the probable number of systems in state s) have the physical meaning. Therefore using a canonical transformation we introduce new variablesnsandθs in a way that the operator
ysys→ys ∂
∂ys →ns (32)
means simply the multiplication by a nonnegative integer ns. Such a canonical transformation will be
ys= Φ(ns)
Φ(ns−1)ehiθs=ehiθsΦ(ns+ 1) Φ(ns) , ys= ∂
∂ys = (ns+ 1)Φ(ns)
Φ(ns+ 1) e−hiθs=e−hiθsnsΦ(ns−1) Φ(ns) ,
(33)
where Φ(n) is an arbitrary function of the integernfor which we demand that atn= 0 it will be equal to unity and for negative nis infinite:6
Φ(0) = 1; 1
Φ(−k) = 0 (k= 1, 2, . . .). (34) The values θs should be considered as canonical variables and the valuesnsas corresponding momenta. Then the meaning ofθs will be:
θs→ −~ i
∂
∂ns, and the operator
e~iθs=e−∂ns∂
will mean “the decrease of the numbernsby unity,” whereas the operator e−~iθs =e∂ns∂
6In the original text the form of the function Φ(n) was related to the type of statistics, whereas in fact it is connected with the normalizing condition; see further the formula (∗). In the present edition this mistake is corrected. (V. Fock, 1957)
means “the increase of the number ns by unity.” The transformation formulae can also be written in the form
ys= Φ(ns)e−∂ns∂ 1 Φ(ns), ys= ∂
∂ys =Φ(ns)
ns! e∂ns∂ ns! Φ(ns).
(33∗)
We must require that the operatorys is conjugate toys. This gives
|Φ(n)|2=n!. Considering Φ(n) to be real, we can put
Φ =p
Γ(n+ 1) =√
n! . (35)
This function evidently satisfies conditions (34). Substituting (35) into (33) we get the transformation used by Dirac, namely,
ys = √
nse~iθs=e~iθs√ ns+ 1, ys =√
ns+ 1e−~iθs =e−~iθs√ ns,
(36)
or
ys = √
nse−∂ns∂ =e−∂ns∂ √ ns+ 1, ys = √
ns+ 1e∂ns∂ =e∂ns∂ √ ns.
(36∗) Besides, we have to show that transformations (33) or (36) are really canonical, i.e., that the next commutation relations between operators ys and ∂
∂ys hold
∂
∂ysyr−yr ∂
∂ys = δrs,
∂
∂ys
∂
∂yr − ∂
∂yr
∂
∂ys = 0, ysyr−yrys = 0.
(37)
Atr=sthe first of these relations follows from equations
∂
∂ys ys=ns+ 1; ys ∂
∂ys =ns, (37∗)
which can be obtained from (33). The other relations are also evidently valid. So transformation (33) is canonical.
§6. Now we have to investigate the transition given by formulae (33) and (36) from the Hilbert space of variables yr to the Dirac space of variablesnr. Let us consider first the case of one variable yr, which we will denote asz.
The function c(n) of an integer numbern in the Dirac space corre- sponds to the functionF(z) of variablezin the Hilbert space. According to the Dirac general theory of representations, the transition fromF(z) toc(n) is realized by the complete system of functionsf(n, z):
F(z) =X
n
c (n)f(n, z). (38)
Before we go further, it should be recalled that expression (35) for Φ(n) was defined from the requirement that both operators (33) are mu- tually conjugated. But the form of the conjugation condition depends on the form of the weight function in the normalizing condition. Therefore the function Φ(n) is connected with the weight function. Namely, the normalizing condition
X
n
n!
|Φ(n)|2 |c(n)|2= 1 (∗) corresponds to the arbitrary Φ(n) and so condition (35) means that the normalizing condition has an ordinary form
X
n
|c(n)|2= 1. (∗∗)
Further in this section we keep Φ(n) arbitrary and put Φ(n) =√ n! only in final formulae.7
We will write equations (33) which define the considered transforma- tion in the form
z → S = Φ(n)
Φ(n−1)e−∂n∂ ,
∂
∂z → T =(n+ 1)Φ(n) Φ(n+ 1) e∂n∂ .
(39)
7The text between the formulae (38) and (39) is added in this edition. (V. Fock, 1957)
They mean that the result of the action of the operatorz(or ∂
∂z) on the left-hand sideF(z) in formula (38) can be expanded in functionsf(n, z), and the expansion coefficientsSc(n) (orT c(n)) will be results of action of operatorS (orT) on the functionc(n). Thus,
zF(z) =X
n
[Sc(n)]f(n, z), (40)
∂F(z)
∂z =X
n
[T c(n)]f(n, z), (41) where, according to (39), the coefficients Sc (n) (or T c (n)) have the following values:
Sc (n) = Φ(n)
Φ(n−1) c(n−1), (42)
T c(n) =(n+ 1)Φ(n)
Φ(n+ 1) c(n+ 1). (43)
Replacingnin (40) byn+ 1 and in (41) byn−1, we can rewrite these formulae in the form:
zF(z) =X
n
Φ(n+ 1)
Φ(n) c(n)f(n+ 1, z), (44)
∂F(z)
∂z =X
n
nΦ(n−1)
Φ(n) c (n)f(n−1, z). (45) On the other hand we can take directly expansion (38) and multiply it byz or take thez-derivative. Then we have
zF(z) =X
n
c (n)zf(n, z), (46)
∂F(z)
∂z =X
n
c(n) f(n, z)
z . (47)
Expressions (44) and (46) and also (45) and (47) should be equal to each other and identically equal relative to the function F(z) and therefore also relative toc (n), i.e., term by term. For that the function f(n, z) must satisfy the following functional equations:
zf(n, z) = Φ(n+ 1)
Φ(n) f(n+ 1, z), (48)
∂f(n, z)
∂z = nΦ(n−1)
Φ(n) f(n−1, z). (49)
Multiplying both parts of (49) byz and expressing the productzf(n− 1, z) according to (48) through f(n, z), we get a differential equation
z∂f(n, z)
∂z =nf(n, z), (50)
the validity of which could also be seen directly from (37∗). Its solution is
f(n, z) =f(n)zn. (51)
Putting (51) in (48) we get
f(n)Φ(n) =f(n+ 1)Φ(n+ 1). (52) Because the quantity (52) does not depend on n, we can simply put it equal to unity.
Thus, we defined the functionf(n, z) up to a factor independent of nandz:
f(n, z) = zn
Φ(n). (53)
For the ordinary normalizing condition (∗∗) we have Φ(n) = √ n! and consequently
f(n, z) = zn
√n!. (53∗)
Transition to many variables proceeds without any difficulties; the eigen- functions are the products of eigenfunctions of a single variable. We write down only the final formula for the canonical transformation of the func- tion Ω (y1, y2, . . .) in the Hilbert space into the functionψ(n1, n2, . . .) in the Dirac space
Ω (y1, y2, . . .) = X
n1,n2,...
ψ(n1, n2, . . .) yn11yn22. . .
Φ(n1)Φ(n2). . . (54) or for the usual normalization of the functionψ
Ω (y1, y2, . . .) = X
n1,n2,...
ψ(n1, n2, . . .) yn11yn22. . .
√n1!√
n2!. . .. (54∗)
§7. The theory of canonical transformation (33) or (36) considered in the previous two sections allows us not only to establish the wave
equation in the Dirac space, but also to get at once its solution if the so- lution in the Hilbert space is known. The transformation of the equation itself is actually not needed; nevertheless, we shall perform it to make easier the comparison with the Dirac formulae.
In formula (21) for the Hamilton function we must replace the op- erators ys and yr = ∂
∂yr by their expressions (33) and (36). Then we get
F =X
s
Kssns+X
r6=s
KrsΦ(nr)Φ(ns)(ns+ 1)
Φ(nr−1)Φ(ns+ 1) e~i(θr−θs) (55) and in the case of usual normalization when Φ(n) =√
n!
F =X
s
Kssns+X
r6=s
Krs√ nr
√ns+ 1e~i(θr−θs). (55∗)
Let us denote the wave function in the Dirac space asψ(n1, n2, . . .).This function satisfies the wave equation
~ i
∂
∂tψ(n1, n2, . . .) +F ψ(n1, n2, . . .) = 0. (56) In an explicit form this equation can be written as
~ i
∂
∂tψ(n1, n2, . . .) +X
s
Kssnsψ(n1, n2, . . .) +
+X
r6=s
KrsΦ(nr)Φ(ns)(ns+ 1)
Φ(nr−1)Φ(ns+ 1)ψ(n1, . . . nr−1, . . . ns+ 1, . . .) = 0. (57) If the functionψ(n1, n2, . . .) is normalized by the formula
X
n1,n2,...
|ψ(n1, n2, . . .)|2= const, (58) then Φ(n) =√
n! and equation (57) gets the form
~ i
∂
∂tψ(n1, n2, . . .) +X
s
Kssnsψ(n1, n2, . . .)+
+X
r6=s
Krs√ nr
√ns+ 1ψ(n1, . . . nr−1, . . . ns+ 1, . . .) = 0. (57∗)
The solution of this equation is already known: ψ(n1, n2, . . .) is the expansion coefficient in (54) or (54∗) if Ω(y1, y2, . . .) satisfies the differ- ential equation (28). This can be checked directly if one inserts (54) or
(54∗) in the differential equation and makes all coefficients zero at all products of any powers ofys; then one gets exactly equation (57) and correspondingly (57∗).
§9.8 Now we are in a position to formulate the statistical problem in a general form and get its solution.
Let us consider two mechanical quantities aandb with operatorsA andBhaving discrete eigenvaluesαsandβs. The probability amplitudes are given at the initial momentt= 0 for the distribution of the systems over the eigenstates of operatorA. They are
ψ0(α)(n1, n2, . . .). (59) We have to calculate the probability amplitudes
ψ(β)t (n1, n2, . . .) (60) for the distribution of systems over the eigenstates of operatorBat time t.
We will get the problem considered by Dirac if operators A and B coincide and if they are equal to the energy operator of the unperturbed system. The solution of the given problem can be found in the following way.
Let us build up, as it was shown in§1, the basic system of functions which are equal to the eigenfunctions of operatorAat t= 0. Consider also the eigenfunctions of operatorB. Using these systems of functions we form the matrixYrs according to (14).
Using the quantities conjugate to the given probability amplitudes (59) and introducing arbitrary parametersg1, g2, . . . ,we form the gen- erating function
Ω0(g1, g2, . . .) = X
n1,n2,...
ψ(α)0 (n1, n2, . . .) gn11g2n2. . .
√n1!√
n2!. . ., (61) where the summation extends over all nonnegative values ofn1, n2, . . ., satisfying the condition
n1+n2+. . .=N (the number of systems). (62) So the function Ω0 is a uniform function of an N-th power relative to parametersgs. Now let us substitute the entering Ω0 parameters gs 8§8 containing an attempt to apply theory to the Fermi statistics is omitted here and§§9, 10 are slightly shortened. (V. Fock, 1957)
with linear forms
gs=X
r
Yrsyr, (63)
whereyrare new parameters. We denote the function ofyr, obtained in this way, by Ω so that
Ω(y1, y2, . . .) = Ω0(g1, g2, . . .). (64) The function Ω satisfies wave equation (27) in the Hilbert space. Let us expand Ω in powers ofysand write down the expansion as
Ω(y1, y2, . . .) = X
n1,n2,...
ψβt (n1, n2, . . .) y1n1yn22. . .
√n1!√
n2!. . .. (65) Then the quantitiesψt(β), conjugate to the expansion coefficients, satisfy the wave equation in the Dirac space and are the wanted probability amplitudes.
§10. Now we consider special cases of the general problem. Let A be the energy operator at timet= 0 and B be the energy operator at timet:
A=H(0), B =H(t). (66)
In this case the functionsψs(q, t) andϕs(q, t) coincide at timet= 0, ψs(q,0) =ϕs(q, t), (67) and the matrix elementYrs turns intoδrsat t= 0, so that att= 0 the quantitiesyscoincide with gs:
ys(0) =gs, Yrs(0) =δrs. (68) The initial values of the expansion coefficientsψ(β)t (n1, n2, . . .) (65) will be the values ψ0(α)(n1, n2, . . .) from formula (61). Hence the method described in §9 allows one to find the solution of the Dirac equation when the initial conditions are given.
Now let us consider the case when the ensemble consists of a single system which is in statesat time t= 0. Then the generating function Ω0 is equal to
Ω0(g1, g2, . . .) =gs (69) and the function Ω(y1, y2, . . .) takes the form
Ω0(y1, y2, . . .) =X
r
Yrs yr. (70)
Comparing this formula with (65), we get ψt³
0,0, . . .1(r),0, . . .´
=Yrs (71)
(the unity in the left-hand side stands on ther-th place).
The square modulus of this value
|ψt|2=|Yrs|2 (72) is the probability that in stater at time t there is one system (in this case the only one). In other words|ψt|2is the transition probability from statesto stater, which coincides with the definition given by Born [3].
As another example, let us consider an ensemble ofN systems; all of them are in the same state att= 0. Then functions Ω0 and Ω will be
Ω0=(gs)N
√N!,
Ω = ÃX
r
Yrsyr
!N
√N! .
(73)
The probability amplitudeψt(n1, n2, . . .) for the distribution (n1, n2, . . .) at timetwill be
ψt(n1, n2, . . .) =
√N!
√n1!√
n2!. . .Y1sn1Y2sn2. . . . (74) The square modulus of this value, which represents the probability of the given distribution, is equal to
|ψt(n1, n2, . . .)|2= N!
n1!n2!. . .|Y1s|2n1|Y2s|2n2. . . . (75) Because the values|Yrs|2give us the probabilities for a single system (see formula (72)), this expression corresponding to the Bose–Einstein statistics coincides with that calculated by the ordinary probability theory.
In conclusion I would like to thank cordially Professor M. Born for his interest in my work and the International Education Board, which made this work possible.