- Email: [email protected]
p-adic dynamical systems
Nuclear Physics B297 (1988) 86-102 North-Holland, Amsterdam
p-ADIC DYNAMICAL SYSTEMS* Peter G.O. FREUND and Mark OLSON
Enrico Fermi Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA Received 3 August 1987
We extend both the classical and quantum mechanics of a harmonic oscillator to cases where the phase space variables are valued in a field F other than the field R of real numbers. Specifically for each prime number p, we consider the cases F - F e, the Galois field with p elements, and more interestingly F - Qp the field of p-adic numbers. The classical mee.hanics is straightforward in a discrete time setting. By contrast, the quantum mechanics is quite a bit richer. We construct explicit and necessarily complex-valued quantum amplitudes. The known additive characters on Fp and Qp play a central role in transmuting the additivity of the classical action into the expected multiplicative properties of the quantum amplitudes. We touch upon further generalizations to other algebraic ("keplerian") dynamical systems and upon phenomenological possibilities. In particular, we propose a p-adic string with complex quantum amplitudes.
1. Motivation
Compact Riemann surfaces figure prominently in string theory where they appear as string world sheets [1]. Compact Riernann surfaces are "algebraic curves" [2] (curves, in that their complex dimension is one while their real dimension is two), and their algebraic geometry has strong arithmetic (number-theoretic) flavor [3]. Some of this arithmetic geometry has already surfaced in discussions of higher loop diagrams in string theory [4]. It is well known that in algebraic geometry it pays to be flexible concerning the nature of the number-field over which the algebraic curves (or varieties) are defined [5]. For instance, by switching to finite fields even the validity of the Riemann hypothesis can be settled (in the affirmative). One may wonder what this flexibility in the choice of number field could yield for dynamical systems. Could one envisage strings over some other field than the field R of real numbers? How about a string over a Galois field, or over the field of p-adic numbers? These questions are interesting, but before we can meaningfully approach them we first will study in detail some much simpler systems for which we have a better control over both the classical and q~antum theories, even as the underlying number-field is being changed. The strings themselves will be discussed * Work supported in part by the NSF grant No. PHY85-21588. 05S0-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishln~ Division)
P.G.O. Freund, M. Olson / p-adic dynamicalsystems
87
at the end of this paper. While this work was in progress, we received a paper by Volovich [6] in which p-adic and Galois strings are being discussed, though in a way not consistent with that of the present paper (see sect. 5). The much simpler problem which we address is that of the classical and quantum mechanics of a harmonic oscillator over a Galois field Fp or over the p-adic field Qp. Dynamical systems over Galois fields have been studied by Hannay and Berry [7] and by Nambu [8]. To do this, it turns out to be both convenient and expeditious to work in discrete time. This is best conceptualized in terms of intermittent observations of the system. One then concentrates on the map K which carries the point in phase-space corresponding to one observation to its position at the next observation. Such an approach is familiar from the study of chaos. It is as if one observed the system's phase-space "stroboscopically" [7]. Under the stroboscopic evolution map K, the phase-space point stays on a constant energy shell. If the equation for this energy shell is algebraic, as is the case for a harmonic oscillator, then it can readily be studied over various number fields. Taken together, the stroboscopic evolution maps corresponding to various "stroboscope settings" reproduce all the information contained in the complete solution of the equations of motion. The maps themselves are readily generalized to other number fields. In a sense this approach bears a close relation to the old ideas of Kepler. To appreciate this, recall that in the classical newtonian point of view one starts from differential equations of motion, themselves derived from an action principle, which are then integrated to yield the general time evolution of the dynamical system. By contrast, the historically earlier, less general, but for special cases just as powerful, keplerian approach, dispenses altogether with equations of motion, but provides instead a full algebraic description of all possible time evolutions of the system. Examples of systems amenable to such a keplerian description are the original Kepler problem of planetary motion, the harmonic oscillator or more recently the instanton problem of non-abelian gauge theory [9]. Just as easily we can provide examples of newtonian systems for which no keplerian presentation exists, e.g. a point particle moving in a Yukawa potential. It appears though that the keplerian systems are the ones that generalize to the p-adic and Galois fields. In the next section we will briefly present some relevant results about p-adic and Galois fields. In sect. 3 we confront the rather simple task of constructing the classical mechanics of the harmonic oscillator over these fields. Sect. 4 is devoted to the, quite a bit less direct, quantization of those systems. In sect. 5 we discuss our results and identify the basic ingredients of a theory of p-adic strings. 2. Galois fields and p-adic fields To make this paper self contained, we present here some basic definitions and well known results on Galois and p-adic fields (impatient experts who are still reading, are invited to go directly on to sect. 3). The simplest Galois field Fp is
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defined as the field of integers modulo the prime p. In detail, the equivalence relation x - y ( m o d p) defines p equivalence classes which can conveniently be chosen as those of the numbers 0,1,2 . . . . . p - 1. For p a prime number, all non-vanishing elements of Fe have a multiplicative inverse so that Fp is a field (e.g. x3 = 5(mod 7)). The field Fe has characteristic p, i.e. for all a, pa =-0 (i.e. pa = 0(mod p)). An additive character on a field Fp is a homomorphism of the additive abelian group F + of F to the multiplicative group C × of the field of complex numbers.
a,b ~ F +,
x ( a + b) = x ( a ) x ( b ) E C.
(2.1)
An additive character on Fp involves the pth roots of unity.
(2.2)
There exist further Galois fields with p" elements (p = prime n ~ Z+), but we shall not confider them in this paper as our primary interest resides in the p-adic numbers. Now to the field Qp of p-adic numbers. These fields are best introduced following Hensel's original reasoning [10]. Hensel started from the analogy between the field of rational numbers and the field R(z) of rational functions of a complex variable z. Each of these is the field of fractions of a ring, the ring of integers in the case of Q, that of polynomials in z in the case of R(z). Each integer can be uniquely decomposed into a product of primes times an overall factor of + 1. In the same vein, each polynomial can be uniquely decomposed in a product of factors of the form ( z - ai) times an overall factor of a complex number. One studies analytic functions in the neighborhood of a point a in the complex plane by expanding it in a Taylor (or Laurent if need be) series around that point, i.e. in a series in terms of powers of the "prime" (z - a~). One can expand the function around the point at infinity in a series in inverse powers of z. It is from these function "elements" that one reconstructs the whole function. Hensel asked for corresponding expansions in the case of rational numbers. Now any rational number can be written, say in basis 10, in the form a,,lO" + a,,_llO "-1
+ an_210 n-2 +
•••
+aolO ° + a_110 -1
+ a_210
-2 +
• • •
which, of course, resembles the expansion of a function around the point at infinity. But imagine that in function theory, this would be all we could do, expand functions at infinity, but nowhere else. The subject would be hardly what it is. What is then the analogue of an expansion in some "prime" (z - a~) for the case of numbers?
P.G.O. Freuna~M. Olson / p-adic dynamical systems
89
Obviously it should be an expansion in increasing powers of some prime p oo
~, anp",
(2.3)
n--k
with k some not necessarily positive finite integer and a . ~ (0,1 ..... p - 1). One can devise a straightforward arithmetic for such formal series "carrying" from left to right (!) rather than from fight to left as in the usual case. Trouble is such series have but formal meaning, they certainly do not converge according to standard Cauchy theory. The individual terms increase in absolute value. But maybe we can define a norm, different from that provided by the absolute value such that the individual terms in the series (2.3) decrease fast according to the new norm. A norm (valuation) II II on a field F is a map from F to the non-negative real numbers such that for x, y ~ F Ilxll = 0 ~ x = 0, Ilxy}l =llxll Ilyll, IIx + y l l ~< Ilxll+llyll.
(2.4)
Can we construct a norm [ Ip such as to give the series (2.3) a meaning? Yes for each prime number p, the p-adic norm [
Ip:
Q~ R
(2.5a)
is defined by
bP~ p p1.
(2.5b)
Here we used the fact that any non-vanishing rational number can be written in the form (a/b)p" where any two of the three integers a, b, p are relatively prime and n is an integer. According to the norm [ [p, the more divisible with p a number is, the smaller it is, the less divisible (the more factors p in the denominator) the larger it is. The p-adic norm I Ip satisfies all the norm constraints (2.4). Indeed I Iv obeys an inequality even stronger than the triangular inequality in (2.2):
Ix + y[p ~
(2.6)
Such a valuation is called non-archimedean. By contrast the Cauchy norm (absolute value) is archimedean it obeys the triangle inequality, but not the stronger inequality
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P.G.O. Freuna[ M. Olson / p-adic dynamical systems
(2.6). By the
p-adic norm the terms in the series (2.3) do decrease as desired. There is one such norm for each prime number p. This raises the important question whether there may be yet other new norms on Q. To settle this question it is convenient to decree two valuations II II and Ill Ill equivalent if there exists a positive real number a such that for any rational number x: IIxll--IIIxlll a. Purely formally let us introduce the notation I [~ for the ordinary absolute value (Cauchy norm). Moreover let us define the trivial norm I I1 by Ixll= 1 i f x ~ 0 and 10Ix = 0 . Then a fundamental theorem of Ostrowski [10] states that: any nontrivial norm on Q is equivalent to some I Ip with p either a prime number, or p = oo. Moreover I le and I Iq are equivalent only if p = q = a prime number or p = q = oo. Now that we have seen that the ordinary Cauchy norm (absolute value) and the p-adic norms are the only conceivable norms (up to equivalence) on Q, we are entitled to ask a crucial question. The Cauchy norm can be used to define (convergent) Cauchy sequences of rational numbers. These sequences will as a rule converge outside Q, which thus gets completed to the field R, the continuum of real numbers. What if instead of the Cauchy norm I I~ we were to use the p-adic norms I Ip in defining the Cauchy sequences? Well, for each p we would end up with a new field Qp the field of p-adic numbers. Each Qp is a continuum. This is most easily seen by noticing that there are as many "p-adic integers" i.e. p-adic numbers whose p-adic norm does not exceed one as there are formal series of the type (2.3) with k = 0. A Cantor diagonal argument then establishes that the p-adic integers form a continuum. Any p-adic non-integer can be written as the ratio of a p-adic integer and a positive integer power of p. The full Qp is thus a continuum. Note that the rational numbers are contained in both R and Qp. In Qp they take the form of the series (2.3) with periodic coefficients a i, just as in R they correspond to base p expansion with repeating digits. The non-rational numbers contained in Qp (their coefficients a, do not repeat) have nothing to do with irrational numbers in R. For instance there is no number corresponding to ~r in Qp. Truncating the series (2.3) at some power n of p will provide an approximation to the corresponding p-adic number. This approximation judged according to the p-adic norm will be better, the larger one chooses n. All rational numbers are inside Qe" In a particular the set of ordinary integers sits inside the set Zp of p-adic integers, densely at that! Even the negative numbers do. For instance in Qs, the number - 1 is represented by 4+4x5+4X52+4x53+4x54+4x5S+.... a "four's complement", in analogy to "one's complement" in base 2. This can be checked by adding it to + 1 and carrying to the right to obtain zero, or by noting that this series is 4(1 + 5 + 25 + 125 + . . . ) = 4/(1 - 5) = - 1.
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P.G.O. Freuna[M. Olson / p-adic dynamicalsystems
Q~, the additive group of Qp, is locally compact and so one can choose [11] a Haar measure dx on Q~. For any a4:0, a ~ Q~ d ( a x ) = I"lpdx is also a Haar measure on Q~'. Just as for Fp, an additive character X exists [11] also for Qe" x(a,p
=e2"°*" k.
Then in particular for k >1O, X(akP k) series
(2.7)
1. Since any p-adic integer x is given by a
=
oo X=
E a k P k, k-O
so that,
x(x)-- 1-[ x(a,p*) -k-0
k-0
(2.8)
1=1,
all p-adic integers have character equal to one. The character of any p-adic number can then be calculated from its series (2.3)
k- -I
oo
X= E akP k, k-m
X(x) =
(
k- -I
)
I-I x(a~pk)=exp 2¢ri Y'. akP k . k-m
~
(2.9)
k-m
The additive character e ipx and the real Haar measure dx on R can be used to define a Fourier transform for functions on R. In the same way we can make use of the character (2.7) and the Haar measure dx to define a Fourier transform for functions on Qp. Let f(x) be a suitably integrable complex valued function on Qp (x ~ Qp ~ F(x) E C). The Fourier transform of f is then given by
f(x) = fQ/(y)x(-xy)dy,
(2.10)
with X given by (2.9). Much of Fourier analysis (Riemann-Lebesgue theorem, convolution-product theorem, etc.) carries over to the p-adic case. Particularly important for us is that the Fourier transform of f(x) -= 1 is a "distribution"
8(x) = i = f ¢ p x ( - x y ) d y ,
(2.11)
ft~ 8(x) f ( x ) dx =f(O),
(2.12)
which has the property
which reveals it as the p-adic Dirac 8-distribution.
P.G.O. Freuna~ M. Olson / p-adic dynamical systems
92
We may add that unlike the real case, where the solution of algebraic equations entails but an extension to the field of complex numbers C, itself already a Cauchy COmplete and algebraically dosed COntinuum, in' the case of Qp the COrresponding extension and COmpletion, I2p involves many more steps. We stop here and warn the reader that this presentation of p-adic numbers is inCOmplete in two respects: many interesting results were not even alluded to, and virtually no proofs were given for the few results mentioned. Yet these are the results we will use in the following sections.
3. The dassical harmonic oscillator over Galois and p-adic fields
The simplest one-dimensional harmonic oscillator hamiltonian over the good old field R of real numbers is 1 H = -f~m[P 2 + (mtoQ) 2]
(3.1)
with m, P, to, Q ~ R. In classical motion, the value E of the hamiltonian function is constant. Introducing the notation ~b= [ R ] ,
R=mtoQ,
F= 2mE,
(3.2)
we have F = ~Tqb.
(3.3)
Integrating Hamilton's equations /~ = toP,
16 -- - t o R ,
(3.4)
one obtains +(t) = K(t)+(O),
(3.5)
with [
tot r ( t ) = [ -COs sin tot
sin tot ]. COs t o t J
(3.6)
The 2 × 2 matrix K ( t ) is orthogonal K X = K -1 '
(3.7a)
P.G.O. Freund, M. Olson / p-adic @namical systems
93
SO that F(t) = F(0), and unimodular det K = 1
(3.7b)
on account of LiouviUe's theorem. It is eq. (3.7a) which clearly identifies the system as a harmonic oscillator. Together the two eqs. (3.7a, b) inform us that K ~ SO(2, R ) .
(3.8)
Now we observe the system stroboscopically at equally spaced time intervals t n = nT. The "stroboscopic" dynamical evolution is governed by the recursion relation qJ.+1 = Kq~.,
(3.9)
with e~, = e # ( n T ) ,
(3.10)
K = K(T).
For a given harmonic oscillator (m and to given) K dials the stroboscope setting, while ¢0 fixes the initial conditions. This stroboscopic presentation carries over directly to a number field F other than R, specifically F = Fp and F = Qp. We keep the basic dynamical equation (3.9) but with the entries of the matrices K and ¢~ valued in F rather than R. For K we again impose eqs. (3.7), so that eq. (3.8) is replaced by K ~ SO(2, F ) ,
(3.11a)
or in detail K-- [ - ba
b] '
a2+b2=l,
a,b~F.
(3.11b)
Various choices of K still correspond to various stroboscopic settings. As an example for p = 7(4 + 25 -- 29 = l(mod 7)) so K--
2 -5
51 2 ] ~ SO(2, F ) .
(3.12)
For F = Fp, the salient feature of the dynamical system defined by eqs. (3.9),(3.11) is that all phase space trajectories are closed, since for any K ~ SO(2, Fp) there exists an integer n ( K , p ) such that K "(x'p) = 1.
(3.13)
Much of the classical and quantum behavior of the system derives from this fact. A
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P.G.O. Freuni M. Olson / p-adic dynamical systems
lot is known about possible values of n(K, p) and the frequencies with which these values arise [7]. When F = Qp, the p-adic field, the set-up (3.9), (3.11) is still appropriate. Truncating the p-adic expansions (eq. (2.3)) as discussed in sect. 2 approximates the p-adic problem by congruences modulo some prime power pn, the full p-adic problem being recovered in the limit n ~ oo. In short both for F=Fp and F= Qp, the classical problem is completely straightforward. By contrast the quantum problem, as will be seen in the next section is quite a bit richer. In preparation for the quantum problem we calculate here the classical action S (Hamilton's principal function) for our systems. Returning to the field R, S expresses the action in terms of the final coordinates (Q2), the initial coordinates (Q1), and the time interval t = t 2 - t x. For a harmonic oscillator S is quadratic S(Q2, Ql, t) = S O+ ~ / Q i Q / .
(3.14a)
The symmetric 2 × 2 matrix S at time intervals t = T is related to the stroboscopic matrix K. For, from Hamilton-Jacobi theory the initial and final momenta Pz and P2 are given by
OS
OS P2 = - -
OQ2
= S21Q1 + S2zQ2,
(3.15)
which when compared with eqs. (3.9) for n = 1 and then making use of the definitions (3.2) and of the eqs. (3.11b) yields
s-- K-~x2L-1 K22
-- b[-1
a "
(3.14b)
Here we assumed the non-triviality of K (K12 b ~ 0). Eqs. (3.14a, b) determine S for time interval T in terms of the matrix K. These expressions carry over bodily to F = Fp and F = Qp and will be of great use in the quantum problem. It may be worthwhile to note that for a very fast stroboscopic action (T ---, 0), the above results can be obtained by direct discretization of the Hamilton eqs. (3.4). Keeping the notations (3.2), (3.10) we introduce the discretization procedure =
,t,(t) -,
+,t,.+,),
~ ( t ) --* q~n+' - q~" T
(3.16)
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P.G.O. Freund~M. Olson / p-adic @namical systems
With the notation that A = ~wT, we find the discretized Hamilton equations.
P,,+,-P.= -A(R,,+R.+t),
(3.17)
which, in terms of ~., can be brought to the form of the recursion relation (3.18a)
~.+i = L # , ~ ,
where 1 [l-A2 L = -1- -+-A -~ - 2 A
2A ] 1 -A 2
(3.18b)
is an orthogonal unimodular 2 X 2 matrix. F_q. (3.18a) has the same form as eq. (3.9) and eq. (3.18b) provides an explicit and well known [12] rational parametrization of the circle a 2 + b 2 = 1 in eq. (3.11b). When transferring this parametrization to the fields F = Fp or F = Qp some care need be exercised. First of all it can happen that 1 + A2 = 0 (in a finite field) and the division in the expression (3.18b) is forbidden. Moreover it can happen that A ---}o0 so that 1/A ~ O, which when one is writing eqs. (3.17) with a factor 1/A on the left-hand side rather than the factor A on the right-hand side, yields the possibility L = - 1 which was missing. In fact, just like K, L can always run over all elements of SO(2, ¥), though for the element - 1 of this group the particular parametrization (3.18b) breaks down and this case is to be recovered directly from Hamilton's equations. In conclusion, the matrices L and K are really one and the same thing. The discretized Hamilton equations can be derived from an action principle obtained by discretization from the hamiltonian action
f/'( PQ. -
H)dt,
(3.19a)
and the addition of a suitable Lagrange multiplier term
1 N~I [(Pn+l + Pn)(Rn+l-
A = 4m---A ~-I
+x [ P : -
+
+ s2)]
R n ) - 12A(Pn +
Pn+l) 2- 21A(Rn+ R.+I) 2] (3.19b)
to be varied with respect to the Lag,range multiplier ~,, the Pi { i = 1. . . . N } and Rj { j = 2 . . . . N - 1 }, but of course not with respect to the initial and final positions Rt, R N. When using the Hamilton equations the action yields as "second order"
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P.G.O. Freund, M. Olson / p-adic dynamical systems
form, the discretization
2T2 n-Z
[(O.+l _
(3.20)
+ 0.):1
of the continuum action
f
- ½mto2Q2dt.
(3.21)
Directly from this action, or less directly from the Hamilton equations one recovers the discretized equation of motion
Rn÷ 1 = 2 1---~iR n - Rn_ 1 .
(3.22)
In conclusion then, one can arrive at the dynamical system (3.9),(3.11) either stroboscopically or directly by discretizing the hamiltonian formalism. We have chosen the method of discrete time because for Fp it is natural, and for Qp it avoids the notorious complications that go with p-adic analysis [10].
4. Quantizafion of the Galois and p-adic harmonic oscillators First consider the Galois field Fp. In this case the problem of quantization has already been solved by Hannay and Berry [7]. On account of F p - Z/pZ tl~ problem can be related to the standard harmonic oscillator. We briefly recapitulate the reasoning of Hannay and Berry and some of their results. This will then permit us to establish some common arithraetic features of the problems over R and Fp, which we then generalize to the p-adic case which is at the center of our interest. The quantum propagator U(q2, ql) for the ordinary harmonic oscillator (over R) is given by the well known expression
iO2S ]x/2 [i2~rS(q2, qt)] U(q2, q,)= hOq, Oqz] exp[ h
'
(4.1)
with S(q 2, qt) given by eq. (3.19) and h Planck's original constant (not h!). Now switching to position Q and momentum P both in Fp, amounts to quantization on a torus in phase space. Wave functions then must be periodic in both their coordinate and momentum representations. This requires that they appear in both these representations as combs of equally spaced 8-functions with periodic coefficients having period equal to p spacings. Specifically if AQ is the period in coordinate
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P.G.O. Freuna[M. Olson / p-adic@namicalsystems
space and A P is the period in momentum space then A PAQ
=
(4.2)
ph.
The spacing between adjacent g-functions in Q-space is then AQ/p, whereas in P-space is A p / p = h / A Q . It is convenient to fix units were A Q = A P = I , which fixes h: (4.3)
h = l/p,
instead of the normal h = 2~rh = 2~r. The p basic Q-space 6-spikes are then located at 1
AQ
P
2AQ
2
,
-
P
=
P
( p - 1)AQ .....
--
p-1 =
P
P
-
-
(4.4)
P
and along the Q-line there are infinitely many periodic repeats of each of these points (at O / p + m , 1 / p + m , 2 / p + m . . . . . p - 1 / p + m , m ~ Z ) . The quantum propagator is then constructed from the familiar propagator over R by lumping one point and all its periodic repeats in an equivalence class and averaging over this class. Such averages reduce to the Gauss sums of number theory. The normaliTation is fixed by unitarity (for details see ref. [7]). Here we wish to briefly consider the instructive special case corresponding to the matrix
Inserting this matrix into eqs. (3.14) and ignoring the additive constant one finds (4.6)
S = - Q1Q2.
At the fundamental points (4.4) in the normalization (4.3), then
S --
h
*~'2 =
*~2
p2 P
p
,
(4.7)
with ~t, ~2 ~ {0,1,2,..., p - 1), so that the exponential in (4.1) takes the form [2~ri
]
exp[7~,~2 j • In fact the whole propagator is then given by [7] Ur,(~2, ~1) =
[plt/2 [2~ri 1 l exp[--~--(-~t~2)l •
(4.8)
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P.G.O. Freund, M. OIson / p-adic dynamical systems
Now let us notice some features of this result. We have
E u;*(4:,
ix)
~2~rp
=
(4.9)
and
E
U(f3,42)U(42,fl)=iS,.+G.o •
(4.10)
Eq. (4.9) expresses the unitarity of Ur. Eq. (4.10) is understood, as well, once one takes into account that classically K1 corresponds to a rotation by 90 °, so that
K12= - I
(4.11)
corresponds to a rotation by 180 °. Viewing Urn(f2, fl) as a p x p matrix with rows and columns labeled by 42 and it, respectively, eq. (4.10) can be rewritten as the matrix equation
UrUK~
=
UK?,
(4.12)
as was to be expected h la Feynman. Actually this claim needs a bit of clarification. How do we know that i8~t+~2,0, which appears in (4.10) is Ucr?)? Were we to attempt a construction along the lines of eqs. (4.5)-(4.8) we would run into the difficulty that eq. (3.14) is ill defined, as it involves division by the off-diagonal element of K~, which however vanishes. Except precisely for this and one other intuitive but singular case one always has
E u ,(f3, 4:)uK(f2, 41) = UK,K(43, 41),
(4.13)
~2~rp
so we could use the result (4.10) as the definition of UK?. In the same way the p x p unit matrix can be used as the definition of Ux_ x. It is trivial to see that these are the only singular cases and that these definitions are precisely those needed to enforce the basic relation (4.13) and the unitarity of the U matrix. The most important feature for us is that, normalization factor aside, the dynamically essential exp(2~i(-4xf2)/p) factor with fl, 42 ~ Fp, is as was already explained in sect. 2 an additive character on Fp. Though the detailed form of S ( - f x 4 2 in this case) changes with the matrix K, in each and every case the propagator is essentially an Fp character. Now we are in a position to generalize to Qp. To quantize over Qp we have to understand what must replace e 2"is/h. To be sure, there is such a thing as a p-adic exponential, but it has none of the properties of a phase factor (none of i, ~r, i¢". . . .
P.G.O. Freuna~114.Olson / p-adic @namicalsystems
99
are in Qp). But there is no compelling reason for the quantum amplitudes to belong to Qp, just because classical quantities did. After all, starting from classical variables valued in R we ended up with quantum amplitudes valued in C. Changing F to Fp classically, we still ended up with quantum amplitudes in C. Why? Because in either case we are dealing with characters of the locally compact abelian group of the number field F over which the classical theory is defined. It is the additive group of Felassiea I that matters because the action valued in Fc1,~i~ is additive. In quantum theory this additivity of the action transmutes into the expected multiplicative properties of quantum amplitudes. That is why characters appear. Characters are valued in C. One may also appreciate the role of characters as crucial in defining Fourier transforms, and thus implementing the uncertainty relations. To quantize the Qp oscillator we have to be given an additive character on Qp. Fortunately, such a thing exists, in sect. 2 we provided such a character (eq. (2.9)). Moreover just as in eqs. (4.9) and (4.10) we were free to sum over intermediate states, so in the p-adic case (just like for R) we have to integrate with the additive Haar measure provided in sect. 2. For the same classical matrix K: (eq. (4.5)), now read as a matrix, with p-adic entries, we can repeat the algebraic steps to get the classical action divide it by h again a p-adic number and get
where Qt and Q2 and therefore - Q I Q 2 / h range over the p-adic numbers and X, the character (2.9) evaluated at these numbers, takes complex values. The quantum amplitudes are complex numbers, yet again! It may be a bit puzzling that the character (2.9) equals one for any p-adic number whose p-adic norm does not exceed one (i.e. for p-adic integers). That this is natural can be seen as follows. The result (4.8), derived from quantization on a phase-space toms, holds even for p replaced by a non-prime number such as p" for instance. Now evaluate (4.8) for ~t~2 an ordinary integer (~1~2 • Z) and p in (4.8) replaced by p". As n increases one will have ever better approximations to the p-adic result (as explained in sect. 2). But as n ~ oo, e x p ( 2 ~ r i ( - ~ l ~ 2 ) / p " ) ~ 1 no matter how ~1~2• Z is chosen. So for all ordinary integers the exponential equals one in the limit. But the ordinary integers sit densely (according to the p-adic norm) in the p-adic integers. So the character is uniquely determined there in accordance with eq. (2.9). Eqs. (4.9), (4.10) now have p-adic counterparts. For instance (using eq. (2.11)):
f
/dO.2×
h
x
(Q2Q,)
=/dQ2x( ) , Q2
h
415,
100
P.G.O. Freund, M. Olson / p-adicdynamicalsystems
as expected. The general case is now dear. Replace - Q1Q2 in (4.14) by the classical action as calculated from the algebraic eqs. (3.14) in terms of the stroboscopic map K. Again one must add the qualification that if the matrix K = K'K' is such that the eq. (3.14) becomes singular, Ux is to be defined from
Ur(Q3, Q1) = f dQ2Ur,,(Q3, Q2)Ur'(Q2, QI).
(4.16)
Again the only singular cases are K = + 1. In principle, with UK known, the p-adic harmonic oscillator is solved. Maybe it bears stressing at this point that the classical dynamical variables are now p-adic numbers. In particular so are space coordinates, or relativistically then also time. We used discrete time, but the integers are contained (densely) in the field Qp. By contrast quantum amplitudes are still complex numbers as in the standard case. 5. Conclusions Here we wish to discuss the picture that emerges from sects. 3 and 4, to see what remains to be done and how these ideas may be further developed. By appealing to a discrete time description we were able to give a very simple presentation of the classical dynamics be it over Fp or Qp. The keplerian nature of the problems (in the sense of sect. 1) allowed us to connect in a simple algebraic manner the discrete time picture with the classical action. This in turn allowed for quantization. In either case the quantum theory returned us to the field of complex numbers, for it is in this field that the all-important characters, which transmute the additivity of the classical action into the expected multiplicativity of the quantum amplitudes, were valued. Obvious questions involve the development of path integrals in the p-adic dynamical systems, and the use of the Feymnan propagator even in our oscillator case to extract say an energy spectrum. Usually this is achieved by leting time go large and imaginary. Being in discrete time, we would first of all have to repeat a map a large number of times, yet unlike the case of theories over R, which allow both unitary (Fourier) and non-unitary (Laplace) characters, over any p-adic field a unitary character has been used in our reasoning, but the corresponding non-unitary character cannot be defined. It remains to be seen how the information contained in the quantum propagator is to be used. One may wonder whether observables are real or p-adic numbers. Take energy for instance. Having used discrete time, and given that the integers are both in R and Qp one may hesitate a bit. But coordinates were p-adic, so relativity already indicates that time should also be p-adic, hence the hamiltonian, the generator of time-translations, should have p-adic eigenvalues. One may want to treat more complicated even if algebraic problems, anharmonic oscillator, strings,... We first wish to comment on Volovich's [6] work on p-adic strings. By analogy with Veneziano's original ansatz, his scattering amplitudes are
P.G.O. Freun~ M. Olson / p-adic @namical systems
101
chosen to be p-adic Beta-functions [10]. These amplitudes are valued in Qp, which is not the home of any characters. This is quite different from our complex amplitudes. Maybe one could imbed Qp in its algebraically closed completion ~2p, but such a quantum theory, if at all possible, would be of a highly novel kind. Short of this possibility it is hard to see how Volovich's proposal could be implemented. Unlike Volovich we want to keep the quantum amplitudes valued in C. For the string we therefore propose to avail ourselves of a different complex-valued betafunction on Qe which has been explicitly constructed [11]. It involves the p-adic integrals of products of complex-valued multiplicative characters on Qp, precisely as in the case of the ordinary Veneziano 4-point function, (the ordinary beta-function). The general N-point Koba-Nielsen ansatz also involves multiple integrals of products of multiplicative characters on R and we also can replicate this p-adically. It is important to explore in detail this p-adic string with complex quantum amplitudes.* If p-adic dynamics bears on reality, one faces the choice of p. Simplistically we may view the problems for all p-adic fields together, and supplemented with the problem for R as all possible completions of a dynamical system originally intended over the rational numbers Q. That the true dynamics should involve the rational rather than the real numbers would be experimentally perfectly conceivable. After all, one measures even the most fundamental parameters (e.g. the fine structure constant) to a finite degree of accuracy which does not preclude their being rational numbers. Choosing a completion according to the Cauchy or p-adic norm is but a convenience, and it may well be that certain arguments would be simplified by a p-adic treatment. In particular, geometrical considerations so heavily biased in favor of real manifolds, allow for unexpected alternative pictures. To actually go as far as choosing a prime and assuming its p-adic field to be fundamental in nature may be a very hard proposition to test, on the basis of not perfectly precise measurements,. for to a finite degree of accuracy a description over the reals may then not be excluded. An approach we find much more promising is based on Weirs adeles [10]. An adele is a sequence (a~, a 2, a 3, a 5. . . . . ap . . . . ) with aoo ~ R , ap ~ Qp for which there exists an integer N such that ap ~ Zp (p-adic integers) for p > N. One can define additive characters on adeles and have a synthesis of all p-adic and oo-adic (real) ways of looking at a dynamical system. We hope to return to this problem elsewhere. We thank Dan Friedan and Lee Brekke for helpful discussions. Discussions with Paul Sally concerning ref. [11] have been extremely useful to us.
* Note added in proof. Beyondits complexquantum amplitudes,such a string deals with an ordinary
real space-time manifold, and this distinguishesit even from the p-adic oscillator discussed here. Such non-archimedean strings have since been constructed [13] and "adelically" connected to the ordinary string [14].
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P.G.O. Freund, M. Olson / p-adic dynamical systems
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13] [14]
P. Nelson, Phys. Reports 149 (1987) 337 H. Farkas and I. Kxa, Riemann surfaces (Springer, Berlin, 1980) G. Cornell and J.H. Silverman, eds., Arithmetic geometry (Springer, Berlin, 1986) Yu I. Martin, Phys. Lett. B172 (1986) 184; L. Alvarez-Gaum~, G. Moore, P. Nelson, and C. Vafa, Phys. Left. B178 (1986) 41 J.H. Silverman, The arithmetic of elliptic curves (Springer, Berlin, 1986) I.V. Volovich, Steklov Math. Inst. preprint (1987) J.M. Hannay and M.V. Berry, Physica 1D (1980) 267 Y. Nambu, preprint EFI 85-52, E.S. Fradkin Festschr., to be published P.G.O. Freund, Proc. 2nd Johns Hopkins Workshop on Current Problems in High Energy Physics, eds. G. Domokos and S. KSvesi-Domokos (Johns Hopkins University, Baltimore, 1978) p. 265 K. Hensel, Theorie der Algebraishen Zahlen (Teubner, Leipzig, 1908); N. Koblitz, p-adic numbers, p-adic analysis, and zeta functions, 2nd ed. (Springer, Berlin, 1984); H. Hasse, Number theory (Springer, Berlin, 1980) I.M. G-errand, M.I. Graev and I.I. Pyatetskii-Shapiro, Representation theory and automorphic functions (Saunders, London, 1966); P.J. Sally and M.H. Taibleson, Acta Math. 116 (1966) 279; M.H. Taibleson, Fourier analysis on local fields (Princeton Univ. Press, Princeton, 1975) N. Koblitz, Introduction to elliptic curves and modular forms (Springer, Berlin, 1984) P.G.O. Freund and M. Olson, Phys. Left. B199, to be published P.G.O. Freund and E. Witten, Phys. Lett. B199, to be published
p-ADIC DYNAMICAL SYSTEMS* Peter G.O. FREUND and Mark OLSON
Enrico Fermi Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA Received 3 August 1987
We extend both the classical and quantum mechanics of a harmonic oscillator to cases where the phase space variables are valued in a field F other than the field R of real numbers. Specifically for each prime number p, we consider the cases F - F e, the Galois field with p elements, and more interestingly F - Qp the field of p-adic numbers. The classical mee.hanics is straightforward in a discrete time setting. By contrast, the quantum mechanics is quite a bit richer. We construct explicit and necessarily complex-valued quantum amplitudes. The known additive characters on Fp and Qp play a central role in transmuting the additivity of the classical action into the expected multiplicative properties of the quantum amplitudes. We touch upon further generalizations to other algebraic ("keplerian") dynamical systems and upon phenomenological possibilities. In particular, we propose a p-adic string with complex quantum amplitudes.
1. Motivation
Compact Riemann surfaces figure prominently in string theory where they appear as string world sheets [1]. Compact Riernann surfaces are "algebraic curves" [2] (curves, in that their complex dimension is one while their real dimension is two), and their algebraic geometry has strong arithmetic (number-theoretic) flavor [3]. Some of this arithmetic geometry has already surfaced in discussions of higher loop diagrams in string theory [4]. It is well known that in algebraic geometry it pays to be flexible concerning the nature of the number-field over which the algebraic curves (or varieties) are defined [5]. For instance, by switching to finite fields even the validity of the Riemann hypothesis can be settled (in the affirmative). One may wonder what this flexibility in the choice of number field could yield for dynamical systems. Could one envisage strings over some other field than the field R of real numbers? How about a string over a Galois field, or over the field of p-adic numbers? These questions are interesting, but before we can meaningfully approach them we first will study in detail some much simpler systems for which we have a better control over both the classical and q~antum theories, even as the underlying number-field is being changed. The strings themselves will be discussed * Work supported in part by the NSF grant No. PHY85-21588. 05S0-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishln~ Division)
P.G.O. Freund, M. Olson / p-adic dynamicalsystems
87
at the end of this paper. While this work was in progress, we received a paper by Volovich [6] in which p-adic and Galois strings are being discussed, though in a way not consistent with that of the present paper (see sect. 5). The much simpler problem which we address is that of the classical and quantum mechanics of a harmonic oscillator over a Galois field Fp or over the p-adic field Qp. Dynamical systems over Galois fields have been studied by Hannay and Berry [7] and by Nambu [8]. To do this, it turns out to be both convenient and expeditious to work in discrete time. This is best conceptualized in terms of intermittent observations of the system. One then concentrates on the map K which carries the point in phase-space corresponding to one observation to its position at the next observation. Such an approach is familiar from the study of chaos. It is as if one observed the system's phase-space "stroboscopically" [7]. Under the stroboscopic evolution map K, the phase-space point stays on a constant energy shell. If the equation for this energy shell is algebraic, as is the case for a harmonic oscillator, then it can readily be studied over various number fields. Taken together, the stroboscopic evolution maps corresponding to various "stroboscope settings" reproduce all the information contained in the complete solution of the equations of motion. The maps themselves are readily generalized to other number fields. In a sense this approach bears a close relation to the old ideas of Kepler. To appreciate this, recall that in the classical newtonian point of view one starts from differential equations of motion, themselves derived from an action principle, which are then integrated to yield the general time evolution of the dynamical system. By contrast, the historically earlier, less general, but for special cases just as powerful, keplerian approach, dispenses altogether with equations of motion, but provides instead a full algebraic description of all possible time evolutions of the system. Examples of systems amenable to such a keplerian description are the original Kepler problem of planetary motion, the harmonic oscillator or more recently the instanton problem of non-abelian gauge theory [9]. Just as easily we can provide examples of newtonian systems for which no keplerian presentation exists, e.g. a point particle moving in a Yukawa potential. It appears though that the keplerian systems are the ones that generalize to the p-adic and Galois fields. In the next section we will briefly present some relevant results about p-adic and Galois fields. In sect. 3 we confront the rather simple task of constructing the classical mechanics of the harmonic oscillator over these fields. Sect. 4 is devoted to the, quite a bit less direct, quantization of those systems. In sect. 5 we discuss our results and identify the basic ingredients of a theory of p-adic strings. 2. Galois fields and p-adic fields To make this paper self contained, we present here some basic definitions and well known results on Galois and p-adic fields (impatient experts who are still reading, are invited to go directly on to sect. 3). The simplest Galois field Fp is
88
P.G.O. Freund, M. Olson / p-adic dynamical systems
defined as the field of integers modulo the prime p. In detail, the equivalence relation x - y ( m o d p) defines p equivalence classes which can conveniently be chosen as those of the numbers 0,1,2 . . . . . p - 1. For p a prime number, all non-vanishing elements of Fe have a multiplicative inverse so that Fp is a field (e.g. x3 = 5(mod 7)). The field Fe has characteristic p, i.e. for all a, pa =-0 (i.e. pa = 0(mod p)). An additive character on a field Fp is a homomorphism of the additive abelian group F + of F to the multiplicative group C × of the field of complex numbers.
a,b ~ F +,
x ( a + b) = x ( a ) x ( b ) E C.
(2.1)
An additive character on Fp involves the pth roots of unity.
(2.2)
There exist further Galois fields with p" elements (p = prime n ~ Z+), but we shall not confider them in this paper as our primary interest resides in the p-adic numbers. Now to the field Qp of p-adic numbers. These fields are best introduced following Hensel's original reasoning [10]. Hensel started from the analogy between the field of rational numbers and the field R(z) of rational functions of a complex variable z. Each of these is the field of fractions of a ring, the ring of integers in the case of Q, that of polynomials in z in the case of R(z). Each integer can be uniquely decomposed into a product of primes times an overall factor of + 1. In the same vein, each polynomial can be uniquely decomposed in a product of factors of the form ( z - ai) times an overall factor of a complex number. One studies analytic functions in the neighborhood of a point a in the complex plane by expanding it in a Taylor (or Laurent if need be) series around that point, i.e. in a series in terms of powers of the "prime" (z - a~). One can expand the function around the point at infinity in a series in inverse powers of z. It is from these function "elements" that one reconstructs the whole function. Hensel asked for corresponding expansions in the case of rational numbers. Now any rational number can be written, say in basis 10, in the form a,,lO" + a,,_llO "-1
+ an_210 n-2 +
•••
+aolO ° + a_110 -1
+ a_210
-2 +
• • •
which, of course, resembles the expansion of a function around the point at infinity. But imagine that in function theory, this would be all we could do, expand functions at infinity, but nowhere else. The subject would be hardly what it is. What is then the analogue of an expansion in some "prime" (z - a~) for the case of numbers?
P.G.O. Freuna~M. Olson / p-adic dynamical systems
89
Obviously it should be an expansion in increasing powers of some prime p oo
~, anp",
(2.3)
n--k
with k some not necessarily positive finite integer and a . ~ (0,1 ..... p - 1). One can devise a straightforward arithmetic for such formal series "carrying" from left to right (!) rather than from fight to left as in the usual case. Trouble is such series have but formal meaning, they certainly do not converge according to standard Cauchy theory. The individual terms increase in absolute value. But maybe we can define a norm, different from that provided by the absolute value such that the individual terms in the series (2.3) decrease fast according to the new norm. A norm (valuation) II II on a field F is a map from F to the non-negative real numbers such that for x, y ~ F Ilxll = 0 ~ x = 0, Ilxy}l =llxll Ilyll, IIx + y l l ~< Ilxll+llyll.
(2.4)
Can we construct a norm [ Ip such as to give the series (2.3) a meaning? Yes for each prime number p, the p-adic norm [
Ip:
Q~ R
(2.5a)
is defined by
bP~ p p1.
(2.5b)
Here we used the fact that any non-vanishing rational number can be written in the form (a/b)p" where any two of the three integers a, b, p are relatively prime and n is an integer. According to the norm [ [p, the more divisible with p a number is, the smaller it is, the less divisible (the more factors p in the denominator) the larger it is. The p-adic norm I Ip satisfies all the norm constraints (2.4). Indeed I Iv obeys an inequality even stronger than the triangular inequality in (2.2):
Ix + y[p ~
(2.6)
Such a valuation is called non-archimedean. By contrast the Cauchy norm (absolute value) is archimedean it obeys the triangle inequality, but not the stronger inequality
90
P.G.O. Freuna[ M. Olson / p-adic dynamical systems
(2.6). By the
p-adic norm the terms in the series (2.3) do decrease as desired. There is one such norm for each prime number p. This raises the important question whether there may be yet other new norms on Q. To settle this question it is convenient to decree two valuations II II and Ill Ill equivalent if there exists a positive real number a such that for any rational number x: IIxll--IIIxlll a. Purely formally let us introduce the notation I [~ for the ordinary absolute value (Cauchy norm). Moreover let us define the trivial norm I I1 by Ixll= 1 i f x ~ 0 and 10Ix = 0 . Then a fundamental theorem of Ostrowski [10] states that: any nontrivial norm on Q is equivalent to some I Ip with p either a prime number, or p = oo. Moreover I le and I Iq are equivalent only if p = q = a prime number or p = q = oo. Now that we have seen that the ordinary Cauchy norm (absolute value) and the p-adic norms are the only conceivable norms (up to equivalence) on Q, we are entitled to ask a crucial question. The Cauchy norm can be used to define (convergent) Cauchy sequences of rational numbers. These sequences will as a rule converge outside Q, which thus gets completed to the field R, the continuum of real numbers. What if instead of the Cauchy norm I I~ we were to use the p-adic norms I Ip in defining the Cauchy sequences? Well, for each p we would end up with a new field Qp the field of p-adic numbers. Each Qp is a continuum. This is most easily seen by noticing that there are as many "p-adic integers" i.e. p-adic numbers whose p-adic norm does not exceed one as there are formal series of the type (2.3) with k = 0. A Cantor diagonal argument then establishes that the p-adic integers form a continuum. Any p-adic non-integer can be written as the ratio of a p-adic integer and a positive integer power of p. The full Qp is thus a continuum. Note that the rational numbers are contained in both R and Qp. In Qp they take the form of the series (2.3) with periodic coefficients a i, just as in R they correspond to base p expansion with repeating digits. The non-rational numbers contained in Qp (their coefficients a, do not repeat) have nothing to do with irrational numbers in R. For instance there is no number corresponding to ~r in Qp. Truncating the series (2.3) at some power n of p will provide an approximation to the corresponding p-adic number. This approximation judged according to the p-adic norm will be better, the larger one chooses n. All rational numbers are inside Qe" In a particular the set of ordinary integers sits inside the set Zp of p-adic integers, densely at that! Even the negative numbers do. For instance in Qs, the number - 1 is represented by 4+4x5+4X52+4x53+4x54+4x5S+.... a "four's complement", in analogy to "one's complement" in base 2. This can be checked by adding it to + 1 and carrying to the right to obtain zero, or by noting that this series is 4(1 + 5 + 25 + 125 + . . . ) = 4/(1 - 5) = - 1.
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P.G.O. Freuna[M. Olson / p-adic dynamicalsystems
Q~, the additive group of Qp, is locally compact and so one can choose [11] a Haar measure dx on Q~. For any a4:0, a ~ Q~ d ( a x ) = I"lpdx is also a Haar measure on Q~'. Just as for Fp, an additive character X exists [11] also for Qe" x(a,p
=e2"°*" k.
Then in particular for k >1O, X(akP k) series
(2.7)
1. Since any p-adic integer x is given by a
=
oo X=
E a k P k, k-O
so that,
x(x)-- 1-[ x(a,p*) -k-0
k-0
(2.8)
1=1,
all p-adic integers have character equal to one. The character of any p-adic number can then be calculated from its series (2.3)
k- -I
oo
X= E akP k, k-m
X(x) =
(
k- -I
)
I-I x(a~pk)=exp 2¢ri Y'. akP k . k-m
~
(2.9)
k-m
The additive character e ipx and the real Haar measure dx on R can be used to define a Fourier transform for functions on R. In the same way we can make use of the character (2.7) and the Haar measure dx to define a Fourier transform for functions on Qp. Let f(x) be a suitably integrable complex valued function on Qp (x ~ Qp ~ F(x) E C). The Fourier transform of f is then given by
f(x) = fQ/(y)x(-xy)dy,
(2.10)
with X given by (2.9). Much of Fourier analysis (Riemann-Lebesgue theorem, convolution-product theorem, etc.) carries over to the p-adic case. Particularly important for us is that the Fourier transform of f(x) -= 1 is a "distribution"
8(x) = i = f ¢ p x ( - x y ) d y ,
(2.11)
ft~ 8(x) f ( x ) dx =f(O),
(2.12)
which has the property
which reveals it as the p-adic Dirac 8-distribution.
P.G.O. Freuna~ M. Olson / p-adic dynamical systems
92
We may add that unlike the real case, where the solution of algebraic equations entails but an extension to the field of complex numbers C, itself already a Cauchy COmplete and algebraically dosed COntinuum, in' the case of Qp the COrresponding extension and COmpletion, I2p involves many more steps. We stop here and warn the reader that this presentation of p-adic numbers is inCOmplete in two respects: many interesting results were not even alluded to, and virtually no proofs were given for the few results mentioned. Yet these are the results we will use in the following sections.
3. The dassical harmonic oscillator over Galois and p-adic fields
The simplest one-dimensional harmonic oscillator hamiltonian over the good old field R of real numbers is 1 H = -f~m[P 2 + (mtoQ) 2]
(3.1)
with m, P, to, Q ~ R. In classical motion, the value E of the hamiltonian function is constant. Introducing the notation ~b= [ R ] ,
R=mtoQ,
F= 2mE,
(3.2)
we have F = ~Tqb.
(3.3)
Integrating Hamilton's equations /~ = toP,
16 -- - t o R ,
(3.4)
one obtains +(t) = K(t)+(O),
(3.5)
with [
tot r ( t ) = [ -COs sin tot
sin tot ]. COs t o t J
(3.6)
The 2 × 2 matrix K ( t ) is orthogonal K X = K -1 '
(3.7a)
P.G.O. Freund, M. Olson / p-adic @namical systems
93
SO that F(t) = F(0), and unimodular det K = 1
(3.7b)
on account of LiouviUe's theorem. It is eq. (3.7a) which clearly identifies the system as a harmonic oscillator. Together the two eqs. (3.7a, b) inform us that K ~ SO(2, R ) .
(3.8)
Now we observe the system stroboscopically at equally spaced time intervals t n = nT. The "stroboscopic" dynamical evolution is governed by the recursion relation qJ.+1 = Kq~.,
(3.9)
with e~, = e # ( n T ) ,
(3.10)
K = K(T).
For a given harmonic oscillator (m and to given) K dials the stroboscope setting, while ¢0 fixes the initial conditions. This stroboscopic presentation carries over directly to a number field F other than R, specifically F = Fp and F = Qp. We keep the basic dynamical equation (3.9) but with the entries of the matrices K and ¢~ valued in F rather than R. For K we again impose eqs. (3.7), so that eq. (3.8) is replaced by K ~ SO(2, F ) ,
(3.11a)
or in detail K-- [ - ba
b] '
a2+b2=l,
a,b~F.
(3.11b)
Various choices of K still correspond to various stroboscopic settings. As an example for p = 7(4 + 25 -- 29 = l(mod 7)) so K--
2 -5
51 2 ] ~ SO(2, F ) .
(3.12)
For F = Fp, the salient feature of the dynamical system defined by eqs. (3.9),(3.11) is that all phase space trajectories are closed, since for any K ~ SO(2, Fp) there exists an integer n ( K , p ) such that K "(x'p) = 1.
(3.13)
Much of the classical and quantum behavior of the system derives from this fact. A
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P.G.O. Freuni M. Olson / p-adic dynamical systems
lot is known about possible values of n(K, p) and the frequencies with which these values arise [7]. When F = Qp, the p-adic field, the set-up (3.9), (3.11) is still appropriate. Truncating the p-adic expansions (eq. (2.3)) as discussed in sect. 2 approximates the p-adic problem by congruences modulo some prime power pn, the full p-adic problem being recovered in the limit n ~ oo. In short both for F=Fp and F= Qp, the classical problem is completely straightforward. By contrast the quantum problem, as will be seen in the next section is quite a bit richer. In preparation for the quantum problem we calculate here the classical action S (Hamilton's principal function) for our systems. Returning to the field R, S expresses the action in terms of the final coordinates (Q2), the initial coordinates (Q1), and the time interval t = t 2 - t x. For a harmonic oscillator S is quadratic S(Q2, Ql, t) = S O+ ~ / Q i Q / .
(3.14a)
The symmetric 2 × 2 matrix S at time intervals t = T is related to the stroboscopic matrix K. For, from Hamilton-Jacobi theory the initial and final momenta Pz and P2 are given by
OS
OS P2 = - -
OQ2
= S21Q1 + S2zQ2,
(3.15)
which when compared with eqs. (3.9) for n = 1 and then making use of the definitions (3.2) and of the eqs. (3.11b) yields
s-- K-~x2L-1 K22
-- b[-1
a "
(3.14b)
Here we assumed the non-triviality of K (K12 b ~ 0). Eqs. (3.14a, b) determine S for time interval T in terms of the matrix K. These expressions carry over bodily to F = Fp and F = Qp and will be of great use in the quantum problem. It may be worthwhile to note that for a very fast stroboscopic action (T ---, 0), the above results can be obtained by direct discretization of the Hamilton eqs. (3.4). Keeping the notations (3.2), (3.10) we introduce the discretization procedure =
,t,(t) -,
+,t,.+,),
~ ( t ) --* q~n+' - q~" T
(3.16)
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P.G.O. Freund~M. Olson / p-adic @namical systems
With the notation that A = ~wT, we find the discretized Hamilton equations.
P,,+,-P.= -A(R,,+R.+t),
(3.17)
which, in terms of ~., can be brought to the form of the recursion relation (3.18a)
~.+i = L # , ~ ,
where 1 [l-A2 L = -1- -+-A -~ - 2 A
2A ] 1 -A 2
(3.18b)
is an orthogonal unimodular 2 X 2 matrix. F_q. (3.18a) has the same form as eq. (3.9) and eq. (3.18b) provides an explicit and well known [12] rational parametrization of the circle a 2 + b 2 = 1 in eq. (3.11b). When transferring this parametrization to the fields F = Fp or F = Qp some care need be exercised. First of all it can happen that 1 + A2 = 0 (in a finite field) and the division in the expression (3.18b) is forbidden. Moreover it can happen that A ---}o0 so that 1/A ~ O, which when one is writing eqs. (3.17) with a factor 1/A on the left-hand side rather than the factor A on the right-hand side, yields the possibility L = - 1 which was missing. In fact, just like K, L can always run over all elements of SO(2, ¥), though for the element - 1 of this group the particular parametrization (3.18b) breaks down and this case is to be recovered directly from Hamilton's equations. In conclusion, the matrices L and K are really one and the same thing. The discretized Hamilton equations can be derived from an action principle obtained by discretization from the hamiltonian action
f/'( PQ. -
H)dt,
(3.19a)
and the addition of a suitable Lagrange multiplier term
1 N~I [(Pn+l + Pn)(Rn+l-
A = 4m---A ~-I
+x [ P : -
+
+ s2)]
R n ) - 12A(Pn +
Pn+l) 2- 21A(Rn+ R.+I) 2] (3.19b)
to be varied with respect to the Lag,range multiplier ~,, the Pi { i = 1. . . . N } and Rj { j = 2 . . . . N - 1 }, but of course not with respect to the initial and final positions Rt, R N. When using the Hamilton equations the action yields as "second order"
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P.G.O. Freund, M. Olson / p-adic dynamical systems
form, the discretization
2T2 n-Z
[(O.+l _
(3.20)
+ 0.):1
of the continuum action
f
- ½mto2Q2dt.
(3.21)
Directly from this action, or less directly from the Hamilton equations one recovers the discretized equation of motion
Rn÷ 1 = 2 1---~iR n - Rn_ 1 .
(3.22)
In conclusion then, one can arrive at the dynamical system (3.9),(3.11) either stroboscopically or directly by discretizing the hamiltonian formalism. We have chosen the method of discrete time because for Fp it is natural, and for Qp it avoids the notorious complications that go with p-adic analysis [10].
4. Quantizafion of the Galois and p-adic harmonic oscillators First consider the Galois field Fp. In this case the problem of quantization has already been solved by Hannay and Berry [7]. On account of F p - Z/pZ tl~ problem can be related to the standard harmonic oscillator. We briefly recapitulate the reasoning of Hannay and Berry and some of their results. This will then permit us to establish some common arithraetic features of the problems over R and Fp, which we then generalize to the p-adic case which is at the center of our interest. The quantum propagator U(q2, ql) for the ordinary harmonic oscillator (over R) is given by the well known expression
iO2S ]x/2 [i2~rS(q2, qt)] U(q2, q,)= hOq, Oqz] exp[ h
'
(4.1)
with S(q 2, qt) given by eq. (3.19) and h Planck's original constant (not h!). Now switching to position Q and momentum P both in Fp, amounts to quantization on a torus in phase space. Wave functions then must be periodic in both their coordinate and momentum representations. This requires that they appear in both these representations as combs of equally spaced 8-functions with periodic coefficients having period equal to p spacings. Specifically if AQ is the period in coordinate
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P.G.O. Freuna[M. Olson / p-adic@namicalsystems
space and A P is the period in momentum space then A PAQ
=
(4.2)
ph.
The spacing between adjacent g-functions in Q-space is then AQ/p, whereas in P-space is A p / p = h / A Q . It is convenient to fix units were A Q = A P = I , which fixes h: (4.3)
h = l/p,
instead of the normal h = 2~rh = 2~r. The p basic Q-space 6-spikes are then located at 1
AQ
P
2AQ
2
,
-
P
=
P
( p - 1)AQ .....
--
p-1 =
P
P
-
-
(4.4)
P
and along the Q-line there are infinitely many periodic repeats of each of these points (at O / p + m , 1 / p + m , 2 / p + m . . . . . p - 1 / p + m , m ~ Z ) . The quantum propagator is then constructed from the familiar propagator over R by lumping one point and all its periodic repeats in an equivalence class and averaging over this class. Such averages reduce to the Gauss sums of number theory. The normaliTation is fixed by unitarity (for details see ref. [7]). Here we wish to briefly consider the instructive special case corresponding to the matrix
Inserting this matrix into eqs. (3.14) and ignoring the additive constant one finds (4.6)
S = - Q1Q2.
At the fundamental points (4.4) in the normalization (4.3), then
S --
h
*~'2 =
*~2
p2 P
p
,
(4.7)
with ~t, ~2 ~ {0,1,2,..., p - 1), so that the exponential in (4.1) takes the form [2~ri
]
exp[7~,~2 j • In fact the whole propagator is then given by [7] Ur,(~2, ~1) =
[plt/2 [2~ri 1 l exp[--~--(-~t~2)l •
(4.8)
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P.G.O. Freund, M. OIson / p-adic dynamical systems
Now let us notice some features of this result. We have
E u;*(4:,
ix)
~2~rp
=
(4.9)
and
E
U(f3,42)U(42,fl)=iS,.+G.o •
(4.10)
Eq. (4.9) expresses the unitarity of Ur. Eq. (4.10) is understood, as well, once one takes into account that classically K1 corresponds to a rotation by 90 °, so that
K12= - I
(4.11)
corresponds to a rotation by 180 °. Viewing Urn(f2, fl) as a p x p matrix with rows and columns labeled by 42 and it, respectively, eq. (4.10) can be rewritten as the matrix equation
UrUK~
=
UK?,
(4.12)
as was to be expected h la Feynman. Actually this claim needs a bit of clarification. How do we know that i8~t+~2,0, which appears in (4.10) is Ucr?)? Were we to attempt a construction along the lines of eqs. (4.5)-(4.8) we would run into the difficulty that eq. (3.14) is ill defined, as it involves division by the off-diagonal element of K~, which however vanishes. Except precisely for this and one other intuitive but singular case one always has
E u ,(f3, 4:)uK(f2, 41) = UK,K(43, 41),
(4.13)
~2~rp
so we could use the result (4.10) as the definition of UK?. In the same way the p x p unit matrix can be used as the definition of Ux_ x. It is trivial to see that these are the only singular cases and that these definitions are precisely those needed to enforce the basic relation (4.13) and the unitarity of the U matrix. The most important feature for us is that, normalization factor aside, the dynamically essential exp(2~i(-4xf2)/p) factor with fl, 42 ~ Fp, is as was already explained in sect. 2 an additive character on Fp. Though the detailed form of S ( - f x 4 2 in this case) changes with the matrix K, in each and every case the propagator is essentially an Fp character. Now we are in a position to generalize to Qp. To quantize over Qp we have to understand what must replace e 2"is/h. To be sure, there is such a thing as a p-adic exponential, but it has none of the properties of a phase factor (none of i, ~r, i¢". . . .
P.G.O. Freuna~114.Olson / p-adic @namicalsystems
99
are in Qp). But there is no compelling reason for the quantum amplitudes to belong to Qp, just because classical quantities did. After all, starting from classical variables valued in R we ended up with quantum amplitudes valued in C. Changing F to Fp classically, we still ended up with quantum amplitudes in C. Why? Because in either case we are dealing with characters of the locally compact abelian group of the number field F over which the classical theory is defined. It is the additive group of Felassiea I that matters because the action valued in Fc1,~i~ is additive. In quantum theory this additivity of the action transmutes into the expected multiplicative properties of quantum amplitudes. That is why characters appear. Characters are valued in C. One may also appreciate the role of characters as crucial in defining Fourier transforms, and thus implementing the uncertainty relations. To quantize the Qp oscillator we have to be given an additive character on Qp. Fortunately, such a thing exists, in sect. 2 we provided such a character (eq. (2.9)). Moreover just as in eqs. (4.9) and (4.10) we were free to sum over intermediate states, so in the p-adic case (just like for R) we have to integrate with the additive Haar measure provided in sect. 2. For the same classical matrix K: (eq. (4.5)), now read as a matrix, with p-adic entries, we can repeat the algebraic steps to get the classical action divide it by h again a p-adic number and get
where Qt and Q2 and therefore - Q I Q 2 / h range over the p-adic numbers and X, the character (2.9) evaluated at these numbers, takes complex values. The quantum amplitudes are complex numbers, yet again! It may be a bit puzzling that the character (2.9) equals one for any p-adic number whose p-adic norm does not exceed one (i.e. for p-adic integers). That this is natural can be seen as follows. The result (4.8), derived from quantization on a phase-space toms, holds even for p replaced by a non-prime number such as p" for instance. Now evaluate (4.8) for ~t~2 an ordinary integer (~1~2 • Z) and p in (4.8) replaced by p". As n increases one will have ever better approximations to the p-adic result (as explained in sect. 2). But as n ~ oo, e x p ( 2 ~ r i ( - ~ l ~ 2 ) / p " ) ~ 1 no matter how ~1~2• Z is chosen. So for all ordinary integers the exponential equals one in the limit. But the ordinary integers sit densely (according to the p-adic norm) in the p-adic integers. So the character is uniquely determined there in accordance with eq. (2.9). Eqs. (4.9), (4.10) now have p-adic counterparts. For instance (using eq. (2.11)):
f
/dO.2×
h
x
(Q2Q,)
=/dQ2x( ) , Q2
h
415,
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P.G.O. Freund, M. Olson / p-adicdynamicalsystems
as expected. The general case is now dear. Replace - Q1Q2 in (4.14) by the classical action as calculated from the algebraic eqs. (3.14) in terms of the stroboscopic map K. Again one must add the qualification that if the matrix K = K'K' is such that the eq. (3.14) becomes singular, Ux is to be defined from
Ur(Q3, Q1) = f dQ2Ur,,(Q3, Q2)Ur'(Q2, QI).
(4.16)
Again the only singular cases are K = + 1. In principle, with UK known, the p-adic harmonic oscillator is solved. Maybe it bears stressing at this point that the classical dynamical variables are now p-adic numbers. In particular so are space coordinates, or relativistically then also time. We used discrete time, but the integers are contained (densely) in the field Qp. By contrast quantum amplitudes are still complex numbers as in the standard case. 5. Conclusions Here we wish to discuss the picture that emerges from sects. 3 and 4, to see what remains to be done and how these ideas may be further developed. By appealing to a discrete time description we were able to give a very simple presentation of the classical dynamics be it over Fp or Qp. The keplerian nature of the problems (in the sense of sect. 1) allowed us to connect in a simple algebraic manner the discrete time picture with the classical action. This in turn allowed for quantization. In either case the quantum theory returned us to the field of complex numbers, for it is in this field that the all-important characters, which transmute the additivity of the classical action into the expected multiplicativity of the quantum amplitudes, were valued. Obvious questions involve the development of path integrals in the p-adic dynamical systems, and the use of the Feymnan propagator even in our oscillator case to extract say an energy spectrum. Usually this is achieved by leting time go large and imaginary. Being in discrete time, we would first of all have to repeat a map a large number of times, yet unlike the case of theories over R, which allow both unitary (Fourier) and non-unitary (Laplace) characters, over any p-adic field a unitary character has been used in our reasoning, but the corresponding non-unitary character cannot be defined. It remains to be seen how the information contained in the quantum propagator is to be used. One may wonder whether observables are real or p-adic numbers. Take energy for instance. Having used discrete time, and given that the integers are both in R and Qp one may hesitate a bit. But coordinates were p-adic, so relativity already indicates that time should also be p-adic, hence the hamiltonian, the generator of time-translations, should have p-adic eigenvalues. One may want to treat more complicated even if algebraic problems, anharmonic oscillator, strings,... We first wish to comment on Volovich's [6] work on p-adic strings. By analogy with Veneziano's original ansatz, his scattering amplitudes are
P.G.O. Freun~ M. Olson / p-adic @namical systems
101
chosen to be p-adic Beta-functions [10]. These amplitudes are valued in Qp, which is not the home of any characters. This is quite different from our complex amplitudes. Maybe one could imbed Qp in its algebraically closed completion ~2p, but such a quantum theory, if at all possible, would be of a highly novel kind. Short of this possibility it is hard to see how Volovich's proposal could be implemented. Unlike Volovich we want to keep the quantum amplitudes valued in C. For the string we therefore propose to avail ourselves of a different complex-valued betafunction on Qe which has been explicitly constructed [11]. It involves the p-adic integrals of products of complex-valued multiplicative characters on Qp, precisely as in the case of the ordinary Veneziano 4-point function, (the ordinary beta-function). The general N-point Koba-Nielsen ansatz also involves multiple integrals of products of multiplicative characters on R and we also can replicate this p-adically. It is important to explore in detail this p-adic string with complex quantum amplitudes.* If p-adic dynamics bears on reality, one faces the choice of p. Simplistically we may view the problems for all p-adic fields together, and supplemented with the problem for R as all possible completions of a dynamical system originally intended over the rational numbers Q. That the true dynamics should involve the rational rather than the real numbers would be experimentally perfectly conceivable. After all, one measures even the most fundamental parameters (e.g. the fine structure constant) to a finite degree of accuracy which does not preclude their being rational numbers. Choosing a completion according to the Cauchy or p-adic norm is but a convenience, and it may well be that certain arguments would be simplified by a p-adic treatment. In particular, geometrical considerations so heavily biased in favor of real manifolds, allow for unexpected alternative pictures. To actually go as far as choosing a prime and assuming its p-adic field to be fundamental in nature may be a very hard proposition to test, on the basis of not perfectly precise measurements,. for to a finite degree of accuracy a description over the reals may then not be excluded. An approach we find much more promising is based on Weirs adeles [10]. An adele is a sequence (a~, a 2, a 3, a 5. . . . . ap . . . . ) with aoo ~ R , ap ~ Qp for which there exists an integer N such that ap ~ Zp (p-adic integers) for p > N. One can define additive characters on adeles and have a synthesis of all p-adic and oo-adic (real) ways of looking at a dynamical system. We hope to return to this problem elsewhere. We thank Dan Friedan and Lee Brekke for helpful discussions. Discussions with Paul Sally concerning ref. [11] have been extremely useful to us.
* Note added in proof. Beyondits complexquantum amplitudes,such a string deals with an ordinary
real space-time manifold, and this distinguishesit even from the p-adic oscillator discussed here. Such non-archimedean strings have since been constructed [13] and "adelically" connected to the ordinary string [14].
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P.G.O. Freund, M. Olson / p-adic dynamical systems
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[12] [13] [14]
P. Nelson, Phys. Reports 149 (1987) 337 H. Farkas and I. Kxa, Riemann surfaces (Springer, Berlin, 1980) G. Cornell and J.H. Silverman, eds., Arithmetic geometry (Springer, Berlin, 1986) Yu I. Martin, Phys. Lett. B172 (1986) 184; L. Alvarez-Gaum~, G. Moore, P. Nelson, and C. Vafa, Phys. Left. B178 (1986) 41 J.H. Silverman, The arithmetic of elliptic curves (Springer, Berlin, 1986) I.V. Volovich, Steklov Math. Inst. preprint (1987) J.M. Hannay and M.V. Berry, Physica 1D (1980) 267 Y. Nambu, preprint EFI 85-52, E.S. Fradkin Festschr., to be published P.G.O. Freund, Proc. 2nd Johns Hopkins Workshop on Current Problems in High Energy Physics, eds. G. Domokos and S. KSvesi-Domokos (Johns Hopkins University, Baltimore, 1978) p. 265 K. Hensel, Theorie der Algebraishen Zahlen (Teubner, Leipzig, 1908); N. Koblitz, p-adic numbers, p-adic analysis, and zeta functions, 2nd ed. (Springer, Berlin, 1984); H. Hasse, Number theory (Springer, Berlin, 1980) I.M. G-errand, M.I. Graev and I.I. Pyatetskii-Shapiro, Representation theory and automorphic functions (Saunders, London, 1966); P.J. Sally and M.H. Taibleson, Acta Math. 116 (1966) 279; M.H. Taibleson, Fourier analysis on local fields (Princeton Univ. Press, Princeton, 1975) N. Koblitz, Introduction to elliptic curves and modular forms (Springer, Berlin, 1984) P.G.O. Freund and M. Olson, Phys. Left. B199, to be published P.G.O. Freund and E. Witten, Phys. Lett. B199, to be published