p-adic quantum-classical analogue of the Heisenberg
uncertainty relations (*)
A. KHRENNIKOV(1)
Mathematical Institute, Ruhr-University - D-44780 Bochum, Germany (ricevuto il 30 Aprile 1996; approvato il 30 Ottobre 1996)
Summary. — We propose a physical interpretation for a p-adic picture of reality.
p-adic numbers describe only measurements with finite accuracy, i.e. all physical measurements. At the same time, real numbers describe measurements with infinite accuracy, i.e. only ideal measurements. Thus, the p-adic picture of reality is more “real” than the real one. Using p-adic analysis, we study the Heisenberg-type uncertainty relations. It is shown that in the p-adic picture these relations are not essentially quantum. We present general uncertainty relations which are valid both in classical and quantum cases.
PACS 02.10 – Logic, set theory and algebra. PACS 11.10 – Field theory.
1. – Introduction
The series of investigations devoted to p-adic strings [1-5] induced a great interest in different p-adic quantum models [6-10] (see, for example, [11] and the next section of the present paper for p-adic numbers). However, this interest has decreased very quickly to practically zero level. The main problem of all p-adic models was their very formal level. Connection with physical reality was problematic. In fact, nobody could show how we might use p-adic numbers to describe real physical phenomena. Moreover, there was the common opinion that the p-adic language might be useful only on the Planck distances (string theory, gravitation and so on).
A new physical interpretation of p-adic models was presented in [12]. In fact, p-adic numbers provide the description of measurements with finite exactness. But these are all physical measurements. At the same time, real numbers provide the description of measurements with infinite exactness. But these are only ideal measurements.
(*) The author of this paper has agreed to not receive the proofs for correction. (1) On leave from Moscow Institute of Electronic Engineering.
Therefore, the p-adic picture of reality is more “real” than the real one. Are there some consequences of the p-adic description which were not evident in the real one? In the present paper we shall show with the aid of p-adic analysis that some essentially quantum effects (from the point of view of ordinary quantum mechanics) are the usual classical effects. These effects are connected only with the exactness of measurement. We shall show that an analogue of the Heisenberg uncertainty principle takes place also for classical particles. Of course, we shall use the p-adic language for classical and quantum particles. In this language infinite exactness of measurement is forbidden.
2. – The mathematical description of a measurement procedure with finite exactness
We always use real numbers to describe measurement procedures both in classical and quantum physics. This real description was used for a very long time (since Netwon’s works). Now, practically nobody pays attention to one sufficiently strange aspect of real formalism. Here we operate with physical quantities which might be measured with infinite exactness. A real number has an infinite number of decimal digits and (at least theoretically) all these digits might be measured. However, every concrete experiment permits only finite exactness of measurement.
Is it possible to include this fixed exactness in a mathematical formalism? We shall try to do this.
What can we get in a measurement S ? Let us choose the unit of measurement 1 and the natural number m (corresponding to the scale of this measurement). As a result of
S we can obtain only quantities of the form
x 4 x2k mk 1 R 1 x21 m 1 x01 R 1 xsm s, (1)
where xj4 0 , 1 , R , m 2 1 are digits in our scale. Denote the set of all such x by
Qm , fin.
Moreover, we could not approach arbitrary finite exactness in S . There exists a fixed number k 4k( S) such that the limit exactness of S equals d( S) 41Omk. It means that we can be sure only of the digit x2kbut the next digit x2(k 1 1 ) is not well defined in S (in this fixed scale).
We wish to create a number system which describes only finite exactness of measure-ment. The set of “physical numbers” Qm , fin has to be the basis of our considerations. First, we are interested in the construction of the field of real numbers R on the basis of Qm , fin. The field R is the completion of Qm , finwith respect to the real metric r(x , y) 4 Nx 2 yN corresponding to the usual absolute value (valuation) N Q N . This metric describes absolute values of physical quantities (with respect to the fixed coordinate system). However, absolute values are not so important in quantum experiments. The exactness of measurement is more important. We define on Qm , fina new valuation corresponding to the exactness d( S).
Set NxNm4m2kfor x4(1) (we assume that x2kc0 ). This is a valuation: 1) NxNmF0 and NxNm4 0 iff x 4 0 ; 2) NxyNmG NxNmNyNm; 3) Nx1yNmG max
(
NxNm, NyNm)
(the strong triangle inequality which implies the ordinary triangle inequality). Setrm(x , y) 4Nx2yNm and complete Qm , fin with respect to this metric (1). Denote this complete metric space by Qm(m-adic numbers). This is a ring with respect to extensions of the usual operations of addition and multiplication. If m 4p is a prime number, then
Qp is a field (of p-adic numbers). The fields of p-adic numbers are more famous in mathematics than the rings of m-adic numbers, see [11]. Of course, it would be better to work in a field than in a ring. However, from the physical point of view it is better to present the general scheme using m-adic numbers.
For any x Qmwe have a unique canonical expansion
(
converging in the NQNm-norm)
of the formx 4x2nO m n
1 R 1 x01 R 1 xkmk1 R , (2)
where xj4 0 , 1 , R , m 2 1 are the “digits” of the m-adic expansion. This expansion contains only a finite number of digits corresponding to negative powers of m. Thus these numbers describe finite exactness of measurement. However, the expansion (2) shows that there is a new moment in the m-adic description which is not present in the real one. There exist quantities described by (2) with an infinite number of digits corresponding to positive powers of m. It is very natural to consider such quantities as infinite quantities (with respect to our fixed unit 1). At the moment, we are not sure that such quantities might be useful in physics. However, there is always the possibility to restrict our attention to finite results.
Now the difference between real and m-adic descriptions of measurement is clear. If the exactness is infinite and the values of all observables are finite, then this is the real-numbers description. If the exactness is finite and the values of the observables may be infinite, then this is the m-adic numbers description.
Here m plays the role of a parameter characterizing a structure of the fixed scale. Different scales are useful for different experiments. However, different m-adic descriptions are (less or more) equivalent from the physical point of view. The exactness
d( S) 41O2k
(the 2-adic description) can be realized as d8( S) 41O3l (of course, not exactly but it suffices for applications). In the same time the rings Qmand Qm8, m c m8, are not isomorphic. Thus, there is no mathematical equivalence.
As usual we define balls in the metric space Qm: Ur(a) 4 ]xQm: rm(x , a) Gr(,
r 4pm
, m 40, 61, 62, R and spheres Ur(a) 4 ]xQm: rm(x , a) 4r(. These sets are closed and open in the same time. Thus our Euclidean intuition does not work in this case. It is important to notice that balls Ur( 0 ) are additive subgroups of Qm.
The balls Ur( 0 ) have a natural interpretation as the restriction of the exactness of measurement by d( S) 41Or. Let xUr( 0 ), r 4ml, then the canonical expansion (2) for
x has the form: x 4a2lO m l
1 R 1 a01 R 1 akmk1 R . For example, if we consider the space M 4Ur3 ( 0 ) 3 R 3 Ur( 0 ), r 4ml, as the model of space, then the exactness is restricted, d 41Oml. If we deny infinite coordinates, then we get a discrete space (also in the classical case).
Remark (A fundamental length). We could propose another physical interpretation
of our formalism. We can consider the fixed finite exactness as a property of space and obtain the formalism of a fundamental length scale and discrete models of space connected with this formalism [13]. What is one of the main advantages of our m-adic
(1) This is the so-called ultrametric, i.e. the strong triangle inequality r (x , y) G max(r (x , z), r (z , y)) holds.
description? This is an additive group structure of our space model with fixed fundamental length and direct connection with rational numbers of which we get in all experiments.
Remark. Our general scheme is based on an idea which is similar to Prugovecki’s
idea [14] to take into account “the individual reading errors” of the individual measurement.
In the following sections we shall use trigonometric functions over the fields of p-adic numbers Qp. These are analytic functions defined by standard series sin x 4
!
(21)nx2 n 11O ( 2 n 1 1 ) ! and cos x 4!
x2 nO ( 2 n) ! . The only difference with the realcase is that these functions are not entirely analytic on Qp. These series converge only on a ball Ur( 0 ) where r 4lp and lp4 1 O p , p c 2 , and l24 1 O 4 . We also need in the calculation of the p-adic valuation of these trigonometric functions: Nsin xNp4 NxNp and Ncos xNp4 1 , see, for example, [11].
3. – An analogue of the Heisenberg uncertainty relations for a quantum-classical harmonic oscillator
We consider the usual harmonic oscillator with mass m and frequency v. However, we use the p-adic description where the coordinate q, momentum p and time t belong to the field of the p-adic numbers Qp. Thus, all these quantities can be measured only with finite exactness. As usual, the Hamilton function has the form: H(q , p) 4 p
2
2 m 1
mv2
2 q
2. We consider m , v Qp , fin. The corresponding Hamilton equations have the form
p8(t) 42mv2q(t),
q8(t) 4p(t)Om .
(3)
It is convenient to choose the initial conditions q( 0 ) 40 and p(0) 4mv. If we restrict our attention to analytic solutions, then we get (as usual) that q(t) 4sin vt, p(t) 4
mv cos vt , where
NvtNpG lp. (4)
Remark. There exist non-analytic smooth solutions, see [11]. But the structure of
these solutions is much more complicated. Further we have: qp 4mv
2 sin 2 vt . Hence, NqpNp4 Nmv O 2 NpNsin 2 vtNp4
NmvNpNvtNp. We can rewrite this equality as the equality for the exactness of measurement of the coordinate q and momentum p of the harmonic oscillator:
d(q) d(p) 4 d(m) d(v)
NvtNp . Now it suffices to use the inequality (4). We get
d(q) d(p) Fd ,
(5)
where d 4d(m) d(v)Olp. Therefore, we always have the uncertainty relation (5) for the coordinate q and momentum p of the harmonic oscillator. We obtain the “quantum”
Heisenberg uncertainty relation if we approach the level of exactness for mass and frequency measurements such that
d 4lph O2 (in the p-scale).
In a general case the exactness of measurement of the initial conditions q( 0 ) and p( 0 ) is also present in the uncertainty relation for the coordinate and momentum. However, we have to notice that these initial conditions are “initial” only with finite exactness of time measurement. As we also have the uncertainty relation
d(t) d(v) F1Olp,
we could not consider time t as measured with infinite exactness. In particular, the zero-time moment is only a moment measured with exactness d(t).
4. – An analogue of the Heisenberg uncertainty relations for a quantum-classical free particle
We consider the motion of a free particle on the p-adic plane Q2p, i.e. the coordinate and momentum of this particle can be measured only with finite exactness. We also suppose that the mass m of the particle is measured with finite exactness in the p-adic scale, m Qp , fin. As usual, the Hamilton function has the form H(p) 4
p2 2 m . The
corresponding Hamilton equations are p8(t) 40, q8(t) 4p(t)Om. If we again restrict our attention to analytic solutions, then we get that p 4p0, q 4q01 (p0O m) t . Hence
d(q) d(p) 4
(
Np0NpNq01 (p0O m) tNp)
21F(
Np0Npmax(
Nq0Np, N(p0O m) tNp))
21 (here we have used the strong triangle inequality). We have the uncertainty relation (5) for the coordinate and momentum of a free particle with d 4d(p0) min(
d(q0),d(p0) d(t) Od(m)
)
. In particular, we obtain the Heisenberg uncertainty relation if we approach the level of exactness d 4hO2.The uncertainty constant d contains the exactness of measurement of the initial conditions q0and p0and the mass m. More interesting is the fact that this constant also contains the exactness d(t) of time measurement. Thus, we cannot consider the uncertainty relation (coordinate)-(momentum) in exactly zero moment of time.
* * *
To propose a physical interpretation to the p-adic numbers description, I had numerous discussions. I would like to thank V. VLADIMIROV, I. VOLOVICH, J. VIGIER, P.
MITTELSTAEDT, B. DRAGOVIC, G. MACKEY, P. FRAMPTON, R. CIANCI, E. BELTRAMETTI, D. FINKELSTEIN, S. ALBEVERIO, A. GRIB for these discussions. This research was realized
on the basis of Alexander von Humboldt Fellowship and visiting professor fellowship at the University of Genova supported by the Italian National Research Council.
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