S Chevillard
email: sylvain.chevillard@ens-lyon.fr

N Revol

Computation of the error functions erf & erfc in arbitrary precision with correct rounding

Keywords: error function, complementary error function, floating-point arithmetic, arbitrary precision, adaptation of the computing precision, correct rounding

In this paper, the computation of erf(x) in arbitrary precision is detailed. A feature of our implementation is correct rounding: the returned result is the exact result (as if it were computed with infinite precision) rounded according to the specified rounding mode. The four rounding modes given in the IEEE-754 standard for floating-point arithmetic are provided. The algorithm that computes the correctly rounded value of erf(x) for any argument x is detailed in this paper. In particular, the choice of the approximation formula, the determination of the order of truncation and of the computing precision are presented. The evaluation formula is written as a partially expanded expression: we explain why it improves the performances in practice. Finally, timings on some experiments are given, and the implementation of the complementary error function erfc is then sketched.

Évaluation des fonctions d'erreur erf et erfc en précision arbitraire avec arrondi correct

Résumé : Dans cet article, nous détaillons comment évaluer la fonction erf en arithmétique flottante à précision arbitraire. Une spécificité de notre implantation est qu'elle retourne le résultat avec arrondi correct : le résultat est égal au résultat exact (le résultat qui serait obtenu si on disposait d'une précision infinie) puis arrondi selon le mode d'arrondi demandé. Les quatre modes d'arrondi de la norme IEEE-754 sont disponibles. Nous présentons un algorithme qui évalue, avec arrondi correct, erf(x) pour tout argument x. En particulier, nous détaillons le choix de la formule d'approximation, la détermination du rang de troncature et de la précision utile pour mener les calculs. La formule utilisée pour l'évaluation a la forme d'une expression partiellement factorisée : nous expliquons en quoi cela améliore les performances en pratique. Nous terminons par des mesures de temps pour quelques expérimentations, puis nous esquissons l'algorithme choisi pour évaluer la fonction complémentaire d'erreur erfc.

Mots-clés : fonction d'erreur, fonction complémentaire d'erreur, arithmétique à virgule flottante, précision arbitraire, arrondi correct, adaptation de la précision de calcul

Introduction

The goal of this work is to compute the error function erf using arbitrary precision floating-point arithmetic and to deliver the correctly rounded result. Correct rounding means that the returned result is the exact result (as if it were computed with infinite precision) rounded according to the specified rounding mode. The four rounding modes given in the IEEE-754 standard for floatingpoint arithmetic, namely rounding to nearest (even), to +∞, to -∞ and to 0 are provided. In particular, having these rounding modes pave the path to an implementation of arbitrary precision interval arithmetic.

The method presented here exhibits two main features. First, the algorithm is completely detailed. Second, remarks on the practical complexity of each kind of operations lead to an evaluation scheme which yields a dramatic reduction of the computing time. The main issue which remains unaddressed is the evaluation of the complementary error function erfc and its use for the evaluation of erf with large arguments.

The error functions are widely used in statistics: the probability that a Gaussian random variable X takes values between m -a and m + a is given by erf a/(σ √ 2) , where m is the mean and σ is the standard deviation of the distribution for X. The error function also appears in the solution of the diffusion equation in specific configurations and in other heat transfer problems. For more information on the error function and the complementary error function, see for instance [START_REF] Lebedev | Special Functions and their Applications[END_REF][START_REF] Temme | Special Functions[END_REF].

Various implementations of the error functions can be found [START_REF] Cody | Rational Chebyshev Approximations for the Error Function[END_REF][START_REF] Cody | Performance evaluation of programs for the error and complementary error functions[END_REF][START_REF] Cody | Algorithm 715: SPECFUN -A Portable FORTRAN Package of Special Function Routines and Test Drivers[END_REF][START_REF] Lozier | Numerical Evaluation of Special Functions[END_REF], but none of the quoted implementations returns the correctly rounded result. Most use the standard floating-point single and double precisions defined by the IEEE-754 standard for floating-point arithmetic [START_REF]IEEE Standard for Binary Floating-Point Arithmetic[END_REF]. Even if the quality of the approximation can be quantified (by an upper bound on the error), none of the above implementations guarantees the correct rounding of the result. Much less implementations are available for arbitrary precision. Maple and Mathematica evaluate the error functions in arbitrary precision. However, Maple suffers from two problems: it does not guarantee the correct rounding of the result, and it is not even clear what the accuracy of the returned result is. Mathematica offers for most functions an indication of the accuracy of the results (based on first order approximations, i.e. not rigorous ones), cf. [START_REF] Sofroniou | Precise numerical computations[END_REF]. The MPFUN package [START_REF] Brent | A Fortran Multiple-Precision Arithmetic Package[END_REF] is better specified [START_REF] Brent | Unrestricted algorithms for elementary and special functions[END_REF]: if 2 -n is the required accuracy, the (relative, except when the result is 0) error between the computed result and the exact value is O(2 -n ). The goal of the DLMF project (Digital Library of Mathematical Functions) is to develop a replacement of Abramowitz and Stegun's Handbook of Mathematical Functions [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF], cf. http://dlmf.nist.gov/. Unfortunately, it will not provide an implementation for the evaluation of special functions. The software accompanying the book [START_REF] Cuyt | Handbook of Continued Fractions for Special Functions[END_REF] is not yet available. The MPFR library (Multiple Precision Floating-point Reliable library, [START_REF] Fousse | MPFR: A Multiple-Precision Binary Floating-Point Library With Correct Rounding[END_REF]) is a library for arbitrary precision floating-point arithmetic with correct rounding. We adopt their motivation for correct rounding: "As a consequence, applications using such a library inherit the same nice properties as programs using IEEE RR n°6465 754-portability, well-defined semantics, ability to design robust programs and prove their correctness-with no significant slowdown on average, with respect to multiple-precision libraries with ill-defined semantics." [START_REF] Fousse | MPFR: A Multiple-Precision Binary Floating-Point Library With Correct Rounding[END_REF]. MPFR includes several special functions (Gamma, Zeta). Simultaneous with our work, the error function has been implemented in MPFR as mpfr_erf. We will compare the implementation of MPFR with ours in Section 4.

We use the MPFR library for our implementation: we use the arbitrary precision arithmetic and algebraic operations and the π constant.

Our algorithm computes an approximation of erf(x) with an absolute error of ε (we will explain later how we choose ε). This gives us a number y such that erf(x) ∈ [y -ε, y + ε]. In order to know if rounding y in a direction r leads to the same result z as rounding the exact value erf(x) in direction r, it suffices to test that y -ε and y + ε rounded in direction r are equal to z. This is achieved by a function can_round.

The scheme of an algorithm that returns the correctly rounded evaluation of a function f on x with rounding mode r and precision p is thus the following:

1. approximate f (x) with computing precision q (the working precision q is determined by the algorithm, it is larger than the target precision p) with error ≤ ε; 2. if can_round(f (x), ε, r, p) then return the correctly rounded value of f (x) in the direction r and precision p; 3. otherwise increase the working precision q, decrease ε and try again.

The error function erf and the complementary error function erfc are introduced in the next Section. Some formulas and properties are given; they are used either to derive an algorithm for the evaluation of erf and erfc or to simplify some cases. Then our algorithm to evaluate erf and to return the correctly rounded result is presented in Section 3. Our implementation is compared to Maple and to the mpfr_erf function of MPFR on some typical examples, in Section 4: for large precisions and small arguments, computing times are significantly reduced. Finally, a list of desirable improvements is given in Conclusion.

The erf and erfc functions 2.1 Definitions

The error function erf and the complementary error function erfc are defined as:

erf : x → 2 √ π x 0 e -t 2 dt, erfc x : x → 1 -erf x = 2 √ π +∞ x e -t 2 dt. ( 1
)
The normalisation factor 2/ √ π ensures that erf defines a probability density function.

Properties

• The erf function is odd: this enables us to consider only nonnegative arguments.

Figure 1: Graph of the erf and erfc functions.

• The value of erf in 0 is erf(0) = 0. Its limit when x tends to +∞ is 1.

• Unfortunately, no properties similar to sin(x + 2π) = sin x or log(x 2 ) = 2 log x hold for the error functions. Such properties prove extremely useful for the evaluation of elementary functions. They allow to reduce the domain to a small interval. Such range reduction is not possible for the erf and erfc functions.

Approximation formulas

Among the possible formulas that can be used to approximate the erf and erfc functions, the book [1, pp. 297-298] gives the integral representations given above (eq. ( 7.1.1) and (7.1.2) in the book), series expansions (eq. (7.1.5) and (7.1.6)) for erf among which (eq. (7.1.5)):

erf x = 2 √ π +∞ n=0 (-1) n x 2n+1 n!(2n + 1) , (2)
an asymptotic expansion for erfc (eq. ( 7.1.23)), a continued fraction for erfc (eq. ( 7.1.14)) and finally an enclosure for erfc (eq. (7.1.13)) that is intensively used by MPFR for large values of x and small precisions.

Algorithm for the erf function

Several approaches could be considered to evaluate erf x where x is a given floating-point number in arbitrary precision. For instance, a numerical quadrature could be performed [START_REF] Gil | Computing special functions by using quadrature rules[END_REF], using the integral representation (1). This implies a large number of evaluations of the exponential function, which is costly: each evaluation implies the evaluation of a sum of the kind given in the power series [START_REF] Brent | A Fortran Multiple-Precision Arithmetic Package[END_REF]. Either this sum is evaluated with correct rounding (as in mpfr_exp) and such an accurate evaluation is costly, or it is approximated less accurately but then the corresponding error must be carefully taken into account in the total error bound.

Determining the best minimax polynomial approximation is not considered when the precision is variable. When the precision is fixed (single or double RR n°6465 IEEE-754 precision), then this polynomial has to be determined only once, but when the precision varies this polynomial has to be recomputed for each precision and this computation is costly. Furthermore, the determination of the approximating polynomial involves the evaluation of the approximated function.

Another solution is to use a Taylor expansion. An advantage of using equation ( 2) is that it involves only rational calculations, whereas the other expansion given by formula (7.1.6) in [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF], with a term e -x 2 in factor of the series, requires one evaluation of the exponential function, which is costly. A second advantage of equation ( 2) is that it is an alternate series and that the remainder of a truncated alternate series is extremely easy to bound: the first neglected term directly provides a bound. On the contrary, the truncation error is not obvious to derive for other expansions. Furthermore, this first neglected term (and thus, the error bound) tends rapidly to 0 as n tends to ∞, at least for small values of x. However, this term of order n can be large when x is large, at least for relatively small values of n. This means that to get a small enough denominator that compensates for the large numerator, it may thus be necessary to truncate after a large number N of terms. Another main drawback of using equation ( 2) is that summing an alternate series usually implies having cancellation. On the opposite, other power series should exhibit no problem of numerical stability, since only positive terms are added. We nevertheless decided upon using equation (2) and we will show how to circumvent this difficulty in Section 3.3.

Several problems must then be solved, in order to derive our algorithm:

• at which number N of terms should we truncate?

• what is an upper bound on the remainder, i.e. on the error due to the truncation? • in which order should the sum be computed in order to minimise the effect of the roundoff errors? • how should the operations be arranged in order to decrease the computing time? • what is an upper bound on these roundoff errors? • what computing precision should be used? • how can the computed result be correctly rounded?

In the following, these questions will receive an answer.

Truncation order and truncation error

Let us denote by a n (x) = (-1) n x 2n+1 / (n!(2n + 1)) and by S N (x) the truncated sum

N n=0 a n (x). The term |a n (x)| is either decreasing with n if x is small enough (x ≤ √ 3
), or it begins by increasing, starting from the value x > √ 3 and then it decreases. The proof of this result can be found in [START_REF] Chevillard | Calcul de la fonction erf en précision arbitraire[END_REF].

The series is an alternating series. Thus, if N is such that the sequence (|a k (x)|) k≥N decreases, it is well known that the truncation error satisfies

∀n ≥ N, erf x - 2 √ π S n (x) ≤ 2 √ π |a n+1 (x)|. Hence, if ε < 2 √ 3/ √ π and if 2/ √ π|a N +1 | ≤ ε, S N (x
) is an approximation of erf(x) with an absolute error less or equal to ε.

Firstly, if x ≥ 1 then the exponent of the floating-point representation of erf x is 0. For smaller x, x is a rough approximation of erf x: we use this value to get an estimation e of the exponent of the floating-point representation of erf x.

If the target precision for the floating-point approximation of erf x is p bits, we truncate the expansion series for erf x at an order N that satisfies 2/ √ π|a N +1 (x)| ≤ 2 e-p-8 , i.e. we choose ε = 2 e-p-8 (note that ε < 2 √ 3/ √ π for p ≥ 9. If p is too small, we use a default value).

The 8 extra bits are for safety, they are expected to enable us to round correctly the computed result. We must also take care of the roundoff errors implied by the floating-point summation: this will be discussed later.

To determine such an order N , we simply compute iteratively the consecutive 8 , using a small computing precision. π and √ π are computed downwards. Then the sequence is computed with rounding upwards. This way, when the process stops, we are sure that 2/ √ π|a k (x)| is actually smaller than 2 e-p-8 . For small target precisions p, we use crude majorations to get an inequality between |a n | and ⌈log 2 n⌉ = α and we try every α in sequence, until n = 2 α -1 satisfies this inequality. Thus, we try less values and we reduce the time of pre-computations.

2/ √ π|a k (x)|, k = 0, 1, • • • until 2/ √ π|a k (x)| < 2 e-p-

Upper bound on the roundoff error of the summands

Using Higham's notation [11, Chapter 3] and intermediate lemmas [START_REF] Chevillard | Calcul de la fonction erf en précision arbitraire[END_REF], we obtained the following bound. Theorem If 5⌊N/2⌋u < 1, with u = ulp(1) = 1 + -1 where 1 + is the smallest floating-point number strictly larger than 1 (using the current computing precision), then the absolute error between 2/ √ πS N (x) and its computed value 2/ √ πS N (x) satisfies

2 √ π S N (x) - 2 √ π S N (x) ≤ 2 √ π e x 2 -1 x γ 5 N 2 (3)
where γ k = (ku)/(1 -ku). More refined majorations (not yet publicly available) enable us to reduce this error. In particular, much more elaborated calculations yield an error bound which is close to the running error bound described in Section 3.6.

Summation order

In [START_REF] Higham | Accuracy and stability of numerical algorithms[END_REF]Chapter 4], N. Higham devotes a complete chapter to the summation problem. The question is: in which order should k summands s 1 , . . . , s k be added in floating-point arithmetic to minimise the roundoff error? When all summands are nonnegative, the recommended order is the increasing order on the added values. In our case, summands are of alternating signs. To reduce the problem to the sum of nonnegative terms, a first idea could be to add odd terms on the one hand, even terms on the other hand, each sum using the increasing order of absolute values. However, this strategy does not work because of heavy cancellations. Our solution consists in grouping the summands by pairs of two consecutive terms: if N is odd,

S N (x) = N n=0 a n (x) = ⌈N/2⌉-1 k=0 (a 2k (x) + a 2k+1 (x)) = x ⌈N/2⌉-1 k=0 x 4k (2k)! 1 4k+1 - x 2 (2k+1)×(4k+1) . ( 4
)
In this case, we do not incur cancellation because the subtracted term is much smaller. Depending on the value of x, the first terms may not be nonnegative, but the last terms are positive and are decreasing.

Our summation algorithm proceeds with k decreasing from ⌈N/2⌉ -1 down to 0. It uses Horner's scheme. The advantages of this strategy, compared to a summation by increasing order of indices, are threefold: the accuracy is good since the computation is done starting from the last (and smallest) terms of a sum of positive non-increasing terms, it is possible to compute a running error bound that is more accurate than the a priori error bound (cf. Section 3.6 and [START_REF] Chevillard | Calcul de la fonction erf en précision arbitraire[END_REF]). Finally the use of the Horner scheme yields a reduced number of operations.

However, for small precisions and small values of x it is preferable to avoid pre-computations and sum directly the terms by increasing values of n.

Summation order: improving the practical performances for large precisions

A careful analysis of the number of operations performed by the Horner scheme applied to the sum given in equation [START_REF] Chevillard | Calcul de la fonction erf en précision arbitraire[END_REF] shows that this evaluation scheme requires, roughly speaking, N/2 multiplications, N additions and 3/2N divisions by an integer. For large precisions, additions are much cheaper than multiplications and divisions. Moreover, even if divisions have in general the same (asymptotic) complexity as multiplications, the situation is not the same for divisions by a (small) integer. In this case the division is much faster than the multiplication, and even the more so as precision becomes large. That is why the cost of the previous method is approximately N/2 times the cost of a high precision multiplication.

We can carry this principle further (described as concurrent series summation in [START_REF] Smith | Algorithm 693; a FORTRAN package for floating-point multipleprecision arithmetic[END_REF]). Let assume L is an integer that divides N . If we group the terms of the series L by L (in consecutive order) in order to form N/L groups, then the overall complexity of the evaluation scheme remains dominated by the number of multiplications. This number is N/L for this new Horner scheme, plus L for the pre-computations of the first L powers of x 2 , used inside each group. The minimum number of operations is 2 √ N and is obtained for L ≃ √ N .

Choice of the computing precision

In order to get a result with p correct bits, one has to perform intermediate computations with q ≥ p bits of precision. We choose to have a roundoff error of the same order ε as the truncation error, i.e. ε = 2 e-p-8 where p is the target precision. Replacing ε by this value and u by 2 1-q in equation ( 3) enables us to deduce the required precision q for the intermediate computations. More precisely, q is the smallest integer such that the roundoff error is less or equal to the truncation error.

More accurate error bounds have been developed both for the case of small argument x and small precision, and for the case of large precision, where the more elaborated Horner scheme presented in Section 3.4 is used. For lack of space, we can only refer the reader to the research report developing these error analyses (to appear).

Rounding the computed result

We have derived an a priori bound on the error between the computed value ŷ and the exact value y = erf x: this bound is 2ε.

It is possible to refine this a priori bound by estimating the error, during the computations. The roundoff error given in ( 3) is overestimated: in particular (3) does not take into account the order of the summation. We adapted the computation of running error bound explained in [START_REF] Higham | Accuracy and stability of numerical algorithms[END_REF]Chapter 5] for Horner's scheme to our problem, to get a new error bound, cf. [START_REF] Chevillard | Calcul de la fonction erf en précision arbitraire[END_REF] for details.

We can then invoke the can_round function with the computed value ŷ that approximates y = erf x, this new error bound and the target precision p to determine if rounding ŷ in the required direction and in precision p is equivalent to rounding y = erf x. If the answer is negative, we restart this process with a larger truncation order N ′ > N and a larger computing precision q ′ > q, cf. Section 3.8.

One remaining question is whether this process will always terminate. It is more generally known as the Table Maker's Dilemma [START_REF] Lefèvre | The Table Maker's Dilemma[END_REF]. To our knowledge, there exists no theorem stating that our process will eventually stop, i.e. stating that there is no floating-point value x (except x = 0) and no precision t such that erf x is exactly a floating-point value using this precision t. To prevent our implementation from looping indefinitely, we may decide to stop it when the required error bound is too small. Obviously, if the user chooses to do this, the implementation will not guarantee correct rounding any more.

How to avoid costly computations when x is large

.. Thus computing erf(x) directly may be a loss of time. For instance, if the required precision is 2k bits, a direct computation will spend a lot of time to compute the first k bits (that are all equal to 1) before computing the interesting last k bits.

A strategy would be to write an algorithm for computing directly erfc(x) = 1 -erf(x) for large values of x. Thus, 2k bits of erf(x) would be obtained by computing approximately k bits of erfc(x) and performing only the subtraction 1 -erfc(x) in high precision.

RR n°6465

We did not implement this strategy yet. See Section 5 where we explain what we are currently doing in this direction.

Adaptation of the computing precision

Kreinovich and Rump [START_REF] Kreinovich | Optimal adaptation of the computing precision[END_REF] established an optimal strategy to automatically adapt the computing precision to the computational needs. The strategy consists in choosing a new precision (and, in our case, a new truncation index) such that the time of the computation using the new precision is twice the time of the previous computation. When the time of the multiplication of two q-bits integers is superlinear (and subquadratic), this yields to multiply by √ 2 the order of truncation and the computing precision. In [START_REF] Kreinovich | Optimal adaptation of the computing precision[END_REF], it is proven that this strategy yields the smallest overhead factor, which is 4, compared to the computational time when the best computing precision were known in advance and used. This is not the strategy adopted by MPFR for the implementation of most of its elementary or special functions: a probabilistic assumption is made, namely that each extra bit of precision divides by 2 the risk of failure. We do not have such probabilistic assumption.

Experimental results

We compare the computing times of Maple, MPFR mpfr_erf and our implementation of erf on some typical values of x (small, medium and large) and various precisions. Results are given in Table 1. Our experiments were conducted on a Intel Pentium 4, 3GHz, 2048KB cache, with Maple 10, gcc 4.1.2, GMP 4.2.1 and MPFR 2.3.1. The precision p is in bits. For Maple (that works with a decimal format) we set the variable Digits to ⌊0.301 • p⌋.

In each line the best time is written in bold. Times are given in milliseconds.

For small precisions and small values of x, our implementation uses basically the same algorithm as MPFR: a direct evaluation of the series without precomputing anything. Hence our timings are close to the timings of MPFR in these cases.

When x > 1.5 and p > 500, our implementation spends some time in computing the order of truncation N and an accurate precision q as described before. This lets us use the technique of grouping terms by blocks of √ N as described in Section 3.4. In these cases our implementation is the most efficient one: the reduction of the number of multiplications really pays, since the cost of each multiplication is costly for large precision.

For large values of x and relatively small precisions, erf x is so close to 1 that the answer can be given almost instantaneously. Our implementation is slower than the other ones because its computations may be somewhat too involved. We plan to implement the function erfc with an efficient algorithm for large values of x. Such an algorithm should enable us to compute erf(x) = 1 -erfc(x) for large values of x by computing erfc(x) with much less precision.

Conclusion and improvements

This work is a first step towards the evaluation of the error functions in arbitrary precisions with correct rounding. Some improvements could yield a better efficiency. A first improvement lies in the complete implementation of erfc as well. We tried the use of continued fraction, because it was easier for us to bound the truncation error, but Brent in [START_REF] Brent | Unrestricted algorithms for elementary and special functions[END_REF] and MPFR prefer the use of the asymptotic series for performance reasons. A second improvement lies in a joint use of erf and erfc, depending on small or large arguments. It is preferable to evaluate erf x when x is small and possibly to compute erfc x as 1 -erf x, since the relevant information lies in erf x. Conversely, it is preferable to evaluate erfc x when x is large. The computing precision, and consequently the computing time, would be reduced.

The problem of infinite loop of the algorithm should also be handled in a more satisfactory way. Any theoretical result on the Table Maker's Dilemma for the error functions is welcome. If it were known that this process halts, even with a huge (theoretical) computing precision, then this problem would vanish. Finally, overflows and underflows occurring during intermediate computations may happen: they should be handled with more care than in our current implementation.

Solving these problems involves a theoretical but also experimental study: thresholds between "small" and "large" precisions and "small" and "'large" arguments must be determined.

  When x is large, the binary representation of erf(x) has the form erf(x) = 0. 11 . . . 11 k b 1 b 2 b 3 . . . The interesting bits are b 1 b 2

Table 1 :

 1 Comparison of Maple, MPFR and our implementation

	p	x	mpfr_erf 2.3.0	Maple 10	our erf
	100	π/100	0.016 ms	0.24 ms 0.010 ms
	100	π	0.10 ms	0.33 ms	0.17 ms
	100 100 1 000	0.25 ms 10 • π < 0.001 ms 2 • π π/100 0.6 ms	0.45 ms 0.015 ms 1.4 ms	0.46 ms 61 ms 0.25 ms
	1 000	π	2.2 ms	2.9 ms	1.0 ms
	1 000 1 000 10 000 π/100 2 • π 10 • π < 0.001 ms 3.6 ms 82 ms	4.1 ms 0.02 ms 120 ms	2.3 ms 92 ms 10 ms
	10 000	π	230 ms	350 ms	27 ms
	10 000 10 000 100 000 π/100 2 • π 10 • π	320 ms 1 100 ms 19 000 ms	480 ms 9 900 ms 33 000 ms	49 ms 500 ms 940 ms
	100 000	π	45 000 ms	81 000 ms 2 000 ms
	100 000 100 000 10 • π 2 • π	56 000 ms 116 000 ms 1 710 000 ms 9 600 ms 101 000 ms 2 500 ms

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Acknowledgements

Our thanks go to C. Q. Lauter for his help in checking the proofs but even more for his help in understanding and improving the practical performances and to the referees of the RNC8 conference for their pertinent remarks.


Elemér E Rosinger
email: eerosinger@hotmail.com

Marie-Louise Nykamp

Must Physics Be Constrained by the Archimedean Axiom ? Relativity and Quanta with Scalars given by Reduced Power Algebras

It is shown that, unknown to nearly everyone, modern theoretical Physics is significantly constrained by the tacit acceptance of the ancient Archimedean Axiom imposed upon Geometry by Euclid more than two millennia ago, an axiom which does not seem to have any modern physical motivation. By freeing oneself of this axiom a large variety of scalar fields and algebras other, and larger than the usual fields R and C of real, respectively complex numbers becomes available for mathematical modelling in theoretical Physics. This paper shows the validity of such modelling in Special Relativity and Quantum Mechanics, namely, in the case of the Lorentz transformations, the Heisenberg Uncertainty and the No Cloning property. The advantages in using such alternative scalars, specifically, given by reduced power algebras or ultrapower fields, are multiple. Among them, one can in a simple and direct way eliminate the so called "infinities in

physics". More generally, one can introduce many levels of precision in theoretical Physics. This is much unlike the present situation when, with the use of R as the only basic scalar field, there can exist only one single level of precision. Also, one can establish a Second Relativity Principle in which the covariance of the equations of Physics and of basic physical phenomena and properties is considered not only with respect to changes of reference frames, but also with changes of algebras or fields of scalars. In this regard, this paper shows that the Lorentz transformations, the Heisenberg Uncertainty and the No Cloning property are indeed covariant with respect to a large variety of scalars given by reduced power algebras, or in particular, ultrapower fields.

"History is written with the feet ..." Ex-Chairman Mao, of the Long March fame ... Science is not done scientifically, since it is mostly done by non-scientists ...

Anonymous

Physics is too important to be left to physicists ...

Anonymous

Is the claim about the validity of the so called "physical intuition" but a present day version of medieval claims about the sacro-sant validity of theological revelations ? Anonymous A "mathematical problem" ? For sometime by now, American mathematicians have decided to hide their date of birth and not to mention it in their academic CV-s. Why ? Amusingly, Hollywood actors and actresses have their birth date easily available on Wikipedia. Can one, therefore, trust American mathematicians ? Why are they so blatantly against transparency ? By the way, Hollywood movies have also for long been hiding the date of their production ...

A bemused non-American mathematician

Part I : Special Relativity in Reduced Power Algebras

A Well Known Usual Deduction of the Lorentz Coordinate Transformations

Here, the Lorentz coordinate transformations, fundamental in Special Relativity, are extended to versions of Special Relativity that are reformulated in terms of scalars in reduced power algebras, instead of the usual real or complex scalars.

The following elementary way to obtain the Lorentz coordinate transformations is well known, [1]. Given two coordinate systems S and S with respective coordinates (x, t) and (x , t ) in which the space x-axis and x -axis are along the same line. We suppose that at time t = t = 0 the origins O and O of the two coordinate systems coincide, thus x = x = 0. Let now S move along thex-axis in the positive direction with the constant velocity v, and let two observers be respectively at O and O .

In that setup, at the initial moment t = t = 0 and when O and O coincide, a light signal is emitted from O. Its propagation within S is then given by (1.1)

x 2 = c 2 t 2
where c > 0 is the velocity of light. Now, in view of the Principle of Constancy of the Velocity of Light, in the coordinate system S the propagation of that light signal is according to

(1.2) x 2 = c 2 t 2
Consequently, one must have

(1.3) x 2 -x 2 = c 2 (t 2 -t 2 )
However, at least for small values of v, when compared with c, we must have (1.4) x = k(c, v)(x -vt)

for some positive k(c, v) ∈ R that does not depend on x, t, x , t , and which in addition is such that (1.5) lim v→0 k(c, v) = 1 since (1.4) and (1.5) are implied by the respective non-relativistic Galilean coordinate transformation. Now in view of the Principle of Relativity of Motion, we can suppose that S is fixed, and S is moving along the x -axis and in the negative direction, with velocity -v. In that case, similar with (1.4), we obtain (1.6) x = k(c, v)(x + vt ) By squaring (1.4) and (1.6), we obtain (1.7)

x 2 + k(c, v) 2 x 2 -2k(c, v)xx = k(c, v) 2 v 2 t 2 (1.8) x 2 + k(c, v) 2 x 2 -2k(c, v)xx = k(c, v) 2 v 2 t 2
thus by subtracting the (1.7) from (1.8), it follows that

(1.9) (x 2 -x 2 )(k(c, v) 2 -1) = k(c, v) 2 v 2 (t 2 -t 2 )
and then in view of (1.3), we obtain

(1.10) c 2 (k(c, v) 2 -1) = k(c, v) 2 v 2 or (1.11) (c 2 -v 2 )k(c, v) 2 = c 2
In this way

(1.12) k(c, v) = c/(c 2 -v 2 ) 1/2 = 1/(1 -v 2 /c 2 ) 1/2
which obviously satisfies (1.5).

The space coordinate Lorentz transformation results now from (1.4) and (1.12), namely

(1.13) x = (x -vt)/(1 -v 2 /c 2 ) 1/2
In order to obtain the time coordinate Lorentz transformation, it will be convenient to proceed in full algebraic detail, and with a special attention to the operations of division and square root involved. For that purpose, we replace x in (1.6) with its value from (1.4). The result is

(1.14) x = k(c, v)(k(c, v)(x -vt) + vt ) = = k(c, v) 2 x -k(c, v) 2 vt + k(c, v)vt or (1.15) k(c, v)vt = k(c, v) 2 vt -(k(c, v) 2 -1)x Thus dividing by k(c, v)v, one has (1.16) t = k(c, v)t -(k(c, v) 2 -1)/(k(c, v)v))x Dividing in (1.11) by c 2 -v 2 , results that (1.17) k(c, v) 2 = c 2 /(c 2 -v 2 )
and then

(1.18) k(c, v) 2 -1 = v 2 /(c 2 -v 2 )
Now (1.16), (1.12) yield the desired time coordinate Lorentz transformation

(1.19) t = (t -vx/c 2 )/(1 -v 2 /c 2 ) 1/2

Extending the Lorentz Coordinate Transformations to Reduced Power Algebras

Let us consider instead of the field R of usual real numbers an arbitrary reduced power algebra A F , see (A.1.4) in the Appendix. In other words, we shall model both space and time with such algebras A F , instead of modelling them with the field R of usual real numbers.

Here it is important to note that, in general, such algebras A F need not be linearly or totally ordered, see (A.4.1) -(A.4.4) in the Appendix. Furthermore, when they are not linearly or totally ordered, that is, when the respective filters F are not ultrafilters, then the corresponding algebras A F need not be one dimensional vector spaces, as is of course the case of R.

It follows that the extension of the Lorentz coordinate transformations to reduced power algebras opens up a rather wide realm, one in which time, as much as each individual coordinate, may be multi-dimensional, and in fact, even infinite dimensional. Speculations regarding the possible meaning of such considerable extensions can, therefore, be diverse and rather numerous. One of them, coming from the multi-dimensionality of time, may be that it could possibly model parallel universes ... And now, let us return to the aimed extension the Lorentz coordinate transformations to arbitrary reduced power algebras.

In this regard, it is sufficient to note that all the algebraic operations in section 1 above, operations leading to the usual Lorentz coordinate transformations in (1.13), (1.19), can automatically be replicated in all the reduced power algebras A F , except when divisions and square roots are involved. Indeed, when divisions are involved in these algebras one has to consider the presence in them of zero divisors and non-invertible elements, see section A.2. in the Appendix. As for square roots, one has to proceed according to section A.5. in the Appendix.

Comments

Why Hold to the Archimedean Axiom ?

It is seldom realized, especially among physicists, that ever since ancient Egypt and the axiomatization of Geometry by Euclid, we keep holding to the Archimedean Axiom. This axiom, in simplest terms, such as of a partially ordered group G, for instance, means the following property

(3.1.1) ∃ u ∈ G, u ≥ 0 : ∀ x ∈ G : ∃ n ∈ N : x ≤ nu
or in other words, there exists a "path length" u, so that every element x in the group can be "overtaken" by a finite number n of "steps" of "length" u. Clearly, if G is the set R of usual real numbers considered with the usual addition, then one can take as u any positive number.

As is known, Geometry in ancient Egypt was important in connection with the yearly flood of the Nile and the subsequent need to redraw the boundaries of agricultural land. And for such a purpose, the Archimedean Axiom is obviously useful.

The question, however, is :

Why hold to that axiom when dealing with such modern and highly non-intuitive theories of Physics, as Special and General Relativity, or Quantum Mechanics and Quantum Field Theory ? Is there any physical type reason in such modern theories for holding to the Archimedean Axiom ?

Indeed, one of the inevitable consequences of the Archimedean Axiom is that "infinity" is not a usual scalar, be it real or complex. Thus all usual algebraic and other operations do rather as a rule break down when reaching "infinity". And this elementary and inevitable fact leads to the long festering problem of the so called "infinities in Physics", a problem which is attempted to be dealt with by various "re-normalization" methods, or by what is an exceedingly complex and so far not yet successful venture, namely, String Theory.

On the other hand, the moment one simply frees oneself from the Archimedean Axiom, and starts to deal with scalars such as those given by various reduced power algebras, the mentioned troubles with "infinity" disappear. Indeed, since the Archimedean Axiom is no longer present in such algebras, these algebras have a rich structure of "infinitesimals" and "infinitely large" scalars, all of which are subjected to the usual algebraic and other operations, just as if they were usual real or complex numbers.

Two Alternatives When Freed From the Archimedean Axiom

The above way the Lorentz Coordinate Transformations have been extended to space-times built upon scalars given by reduced power algebras may at first seem to be both trivial and without interest. And the same appearance may arise with the extension to such space-times of the Heisenberg Uncertainty and No-Cloning, in [10], respectively, [11].

Here however, one should note the following.

First, even the multiplication in such reduced power algebras is no longer trivial. Indeed, such algebras can have zero divisors, see section A.2. in the Appendix. Consequently, it may easily happen that, although c, v, x, t, k(c, v) = 0, we will nevertheless have the products in which such quantities appear, and the respective products vanish, contrary to what happens in the usual case when scalars given by real numbers are employed. And clearly, such a vanishing of certain products may invalidate subsequent formulas, or at best, give them a different meaning from the usual one. Also, mathematical expressions in various theories of Physics can contain operations other than mere multiplication, and such operations can have new properties and meanings, when performed in reduced power algebras.

Therefore, here, we may obviously face two rather different alternatives, namely

• the new properties and meanings in reduced power algebras do not correspond to any possible physical meaning, or on the contrary

• such new properties and meanings which appear in reduced power algebras may possibly correspond to not yet explored physical realities.

We shall in the sequel mention several possible such new physical interpretations, if not in fact, possible realities.

Increased and Decreased Precision in Measurements

As a general issue, relating not only to Relativity or the Quanta, the presence of infinitesimal and infinitely large scalars in reduced power algebras may correspond to a new possibility of having no less than two radically different kind of measurements when it comes to their relative precision.

Namely, one has an increased precision in measurement, when measurement is done in terms of usual finite scalars, and one obtains as result some infinitesimal scalar in such algebras.

Alternatively, the presence of infinitely large scalars in such algebras may simply indicate that they were obtained in terms of finite scalars, and thus are but the result of a measurement with decreased precision.

In this regard, we can therefore have the following relative situations

• infinitesimal scalars are the result of increased precision measurements done in terms of finite or infinite scalars,

• finite scalars are the result of increased precision measurements done in terms of infinite scalars,

• finite or infinitely large scalars are the result of decreased precision measurements done in terms of infinitesimal scalars,

• infinitely large scalars are the result of decreased precision measurements done in terms of infinitesimal or finite scalars.

and surprisingly, one can also have the following relative situations

• infinitesimal scalars are the result of increased precision measurements done in terms of some less infinitesimal scalars,

• infinitesimal scalars are the result of decreased precision measurements done in terms of some more infinitesimal scalars,

• infinitely large scalars are the result of increased precision measurements done in terms of some more infinitely large scalars,

• infinitely large scalars are the result of decreased precision measurements done in terms of some less infinitely large scalars.

Indeed, one of the basic features of reduced power algebras is precisely their complicated and rich self-similar structure which distinguishes not only between infinitesimal, finite and infinitely large scalars, but also within the infinitely small scalars themselves, and similarly, within the infinitely large scalars. Specifically, infinitesimal scalars can be infinitely smaller, or on the contrary, infinitely larger than other infinitesimals. And similarly, infinitely large scalars can be infinitely smaller, or on the contrary, infinitely larger than other infinitely large scalars.

Here, however, we can note that such a possible interpretation of increased, or decreased precision which is relative, is in fact not new. Indeed, in terms of usual scalars, be they real or complex, there is a marked dichotomy between finite scalars, and on the other hand, the so called "infinities" which may on occasion arise from operations with finite scalars. And such simple "formulas" like ∞ + 1 = ∞, are in fact expressing that fact. Namely, on one hand, from the point of view of "infinity", the finite number 1 has such an increased precision as to be irrelevant with respect to addition, while on the other hand, from the point of view of the finite number 1, the "infinity" has such a decreased precision as to alter completely the result when involved in addition.

The Issue of Universal Constants

Given the above possibilities in interpretation leading to relative precision measurement -be it as such an increased or a decreased oneone can reconsider the status of certain universal physical constants, such as for instance, the Planck constant h and the constant c giving the velocity of light in vacuum. Indeed, when considered from our everyday macroscopic experience, h is supposed to be unusually small, while on the contrary, c is very large. Consequently, one may see h as a sort of "infinitesimal", while c then looks like "infinitely large". The fact is that, within reduced power algebras, such an alternative view of h and c is possible. Therefore, one may find it appropriate to explore the possible physical meaning, or otherwise, that may possibly be associated with such an interpretation.

(A.4.3) For every x ∈ R Λ , the four alternatives (A.2.3.1) -(A.2.3.4) reduce to the following two, namely, (A.2.3.2), (A.2.3.3), that is :

Z(x) ∈ F and Λ \ Z(x) / ∈ F Z(x) / ∈ F and Λ \ Z(x) ∈ F (A.4.4) The partial order ≤ F in (A.1.6
) is a linear, or total order on the reduced power field A F

A.5. Exponential Functions

In view of (A.1.9), one can obviously define the exponentiation (A.5.1)

A + F × A + F ((x) F , (y) F ) -→ (z) F = ((x) F ) ((y) F ) ∈ A + F by (A.5.2) z(λ) = (x(λ)) (y(λ)) , λ ∈ I where I ∈ F is such that (A.5.3) x(λ), y(λ) ≥ 0, λ ∈ I
Part II : Heisenberg Uncertainty in Reduced Power Algebras

Preliminaries

The Heisenberg uncertainty relation is known to be obtainable by a purely mathematical argument. Based on that fact, here it is shown that the Heisenberg uncertainty relation remains valid when Quantum Mechanics is re-formulated within far wider frameworks of scalars, namely, within one or the other of the infinitely many reduced power algebras which can replace the usual real numbers R, or complex numbers C. A major advantage of such a re-formulation is, among others, the disappearance of the well known and hard to deal with problem of the so called "infinities in Physics". The use of reduced power algebras also opens up a foundational question about the role, and in fact, about the very meaning and existence, of fundamental constants in Physics, such as Planck's constant h. A role, meaning, and existence which may, or on the contrary, may not be so objective as to be independent of the scalars used, be they the usual real numbers R, complex numbers C, or scalars given by any of the infinitely many reduced power algebras, algebras which can so easily be constructed and used.

A remarkable feature of the Heisenberg uncertainty relation is that it can be obtained following a purely mathematical argument of a rather simple statistical nature, [1, pp. 67-70]. Based on that fact, here we shall show that the Heisenberg uncertainty relation remains valid when Quantum Mechanics is re-formulated within what appears to be a far more wide and appropriate framework of scalars, namely, those in any of the infinitely many algebras which belong to the class of reduced power algebras, [3][4][5][6][7][8][9][10].

As argued in [3][4][5][6][7][8][9][10], there are a number of important advantages in re-formulating the whole of Physics in terms of scalars given by reduced power algebras. Related to Quantum Theory, and specifically, to Quantum Field Theory, one of the major advantages of such a reformulation is the complete and automatic disappearance of the well known and hard to deal with problem of the so called "infinities in Physics".

In this regard, let us recall that in [3][4][5][6][7][8][9][10] it was shown how to construct in a simple way as reduced powers a large class of algebras which extend the field R of usual real numbers, or alternatively, the field C of usual complex numbers. A remarkable feature of these reduced power algebras is that they contain infinitesimal, as well as infinitely large elements, consequently, these algebras are non-Archimedean. Some of these algebras are in fact fields. Also, among them is the field * R of nonstandard real numbers.

In [3][4][5][6][7][8][9][10] it was suggested and argued that much of present day Physics should be re-formulated in terms of such reduced power algebras, one of the main reasons for that being the considerably increased richness and complexity of their non-Archimedean self-similar mathematical structures, as opposed to the much simpler structures imposed on Physics by the Archimedean field R of the usual real numbers, and by the structures built upon it, such as the field C of complex numbers, various finite or infinite dimensional manifolds, Hilbert spaces, and so on. And in this regard, it was argued that one of the main advantages of such a re-formulation would be the automatic disappearance of the difficulties related to the so called "infinities in Physics", as a result of the presence of infinitely large elements in the reduced power algebras.

It was also pointed out that the present limitation to the exclusive use of scalars, vectors, etc., which belong to Archimedean mathematical structures is the result not of absolutely any kind of conscious and competent choice in Physics, but on the contrary, of the millennia long perpetuation of a mere historical accident, namely that in ancient Egypt the development of Geometry chose the Archimedean route, due to specific practical needs at those times, needs hardly at all related to those of modern Physics.

However, as not seldom happens in human affairs, accidentally acquired habits can become second nature. This may explain, even if not excuse as well, why for more than two millennia by now we have been so happily wallowing in the ancient Egyptian bondage, or rather slavery of Archimedean space-time structures ...

For convenience, we shall recall in a particular case the construction, [3][4][5][6][7][8][9][10], of reduced power algebras. Given any filter F on N, we define

(1.1) I F = { v = (v n ) n∈N ∈ R N | { n ∈ N | v n = 0 } ∈ F }
which is a proper ideal in the algebra R N . Thus we obtain the reduced power algebra associated to F as the quotient algebra

(1.2) R F = R N /I F
Furthermore, this algebra which is commutative, is also a strict extension of the field R of the usual real numbers, according to the embedding of algebras

(1.3) R x -→ (x, x, x, . . .) + I F ∈ R F = R N /I F
In a similar manner one can obtain reduced power algebras extending the field C of the usual complex numbers. Namely, let us denote by

(1.4) J F = { w = (w n ) n∈N ∈ C N | { n ∈ N | w n = 0 } ∈ F }
which is a proper ideal in the algebra C N . Thus we obtain the reduced power algebra associated to F as the quotient algebra

(1.5) C F = C N /J F
Furthermore, this algebra which is commutative, is also a strict extension of the field C of the usual complex numbers, according to the embedding of algebras

(1.6) C z -→ (z, z, z, . . .) + I F ∈ C F = C N /J F
We now establish a natural connection between the algebras R F and C F .

In this regard, we note the following connection between the ideals I F and J F . Namely

(1.7)

w = (w n = u n + iv n ) n∈N ∈ J F ⇐⇒ ⇐⇒ u = (u n ) n∈N , v = (v n ) n∈N ∈ I F where u n , v n ∈ R.
It follows that we have the algebra homomorphisms (1.8)

Re : C F w = (w n = u n + iv n ) n∈N + J F -→ -→ u = (u n ) n∈N + I F ∈ R F (1.9) Im : C F w = (w n = u n + iv n ) n∈N + J F -→ -→ v = (v n ) n∈N + I F ∈ R F
as well as the algebra embeddings

(1.10) R F u = (u n ) n∈N + I F -→ u = (u n ) n∈N + J F ∈ C F (1.11) R F v = (v n ) n∈N + I F -→ iv = (iv n ) n∈N + J F ∈ C F
Let us also define the surjective linear mapping (1.12)

C F w = (w n = u n + iv n ) n∈N + J F -→ -→ w = (w n = u n -iv n ) n∈N + J F ∈ C F
As a consequence, we obtain

(1.13) w = (w n = u n + iv n ) n∈N + J F ∈ C F , w = w =⇒ w ∈ R F
Lastly, we can define the absolute value on C F , by the mapping (1.14)

C F z = (w n = u n + iv n ) n∈N + J F -→ -→ |z| = (|w n | = (u 2 n + v 2 n )) n∈N + I F ∈ R F Let us denote (1.15) R + F = { u = (u n ) n∈N + I F ∈ R F | { n ∈ N | u n ≥ 0 } ∈ F }
then we obtain the surjective mapping

(1.16) C F z -→ |z| ∈ R + F and for z ∈ C F , we have (1.17) |z| = 0 ⇐⇒ z = 0
Now, in view of (1.8), (1.9), (1.14), we have for z ∈ C F the relations

(1.18) | Re z |, | Im z | ≤ | z |
For further convenience, we shall consider a quantum configuration space which is one dimensional.

Here however, there are two ways to proceed.

The simpler one is to model the one dimensional configuration space by the usual R, in which case the wave functions will be given by

(1.19) ψ : R -→ C F
This means that the only difference with the usual quantum mechanical setup is that, this time, the wave functions can take values in the reduced power algebra extension C F of C.

Alternatively, one can be more consistent in the re-formulation of Physics in terms of reduced power algebras, and model not only the values of the wave functions, but also their one dimensional configuration space variables with the reduced power algebra R F which is an extension of R. Thus in this second case, the wave functions would be

(1.20) ψ : R U -→ C F
where U is an ultrafilter on N.

The first of these two alternatives will be developed in the next section.

The second and yet more general alternative is treated elsewhere.

An Extension of the Heisenberg Uncertainty

In order to avoid unnecessary technical complications concerning the integrations related to wave functions ψ in (1.19), we shall only consider those of them which are of the following particular step function type

(2.1) ψ = 1≤h≤m γ h H h
where m ≥ 1, and γ h ∈ C F , while H h : R -→ {0, 1} are step functions such that H h (x) = 1, when x ∈ I h , and H h (x) = 0, when x / ∈ I h . Here I h = [a h-1 , a h ) ⊂ R are usual intervals, where ∞ ≤ a 0 < a 1 < a 2 ≤ . . . < a m ≤ ∞ are usual real numbers, with a 0 and a m possibly minus and plus infinity, respectively.

The set of all such functions wave functions ψ in (2.1) is denoted by

(2.2) S F (R)
and this set replaces the usual Hilbert space L 2 (R) of complex valued square integrable wave functions ψ defined on the configuration space R of a one dimensional quantum system.

We note here one of the major advantages in the above use of reduced power algebras. Namely, each of the wave functions ψ ∈ S F (R) can easily be integrated on the whole of R regardless of the possible infinite length of some of the intervals I h , or of the infinite value of some of the γ h . Indeed, owing to the reduced power algebra structure of C F , one simply obtains in C F the algebraically perfectly well defined value

(2.3) R ψ(x)dx = 1≤h≤m (a h -a h-1 )γ h ∈ C F
which is always a well defined element in C F , and thus available in a correct and rigorous manner for all the algebraic operations in the algebra C F , even if that value may turn out to be an infinitesimal, finite, or an infinitely large element in C F . In this way there is no need to impose any integrability type conditions on the wave functions ψ ∈ S F (R), much unlike in the usual case, where the square integrability condition R |ψ(x)| 2 dx < ∞ is required, since within the usual Archimedean framework of C, or R, one cannot perform most of the usual algebraic operations with infinitely large quantities. Now it follows easily that S F (R) is a vector space over C, and in fact, it is a commutative algebra over C. Furthermore, one can define on it the extension of the usual scalar product given by (2.4) < ψ, χ > = R ψ(x)χ(x)dx ∈ C F for all ψ, χ ∈ S F (R). This extended scalar product has the following properties :

(2.5) It is linear over C F , therefore also over C, in the second argument.

(

2.6) < χ, ψ > = < ψ, χ >, ψ, χ ∈ S F (R) (2.7) < ψ, ψ > ∈ R + F , ψ ∈ S F (R)
and for ψ ∈ S F (R), one has

(2.8) < ψ, ψ > = 0 ⇐⇒ ψ = 0 ∈ S F (R)
Also, we have the extension of the classical Schwartz inequality

(2.9) | < ψ, χ > | ≤ < ψ, ψ > 1/2 < χ, χ > 1/2 , ψ, χ ∈ S F (R)
Lastly, we can consider the set (2.10) L(S F (R)) of all linear operators A : S F (R) -→ S F (R). Such an operator will be called Hermitian, if and only if

(2.11) < Aψ, χ > = < ψ, Aχ >, ψ, χ ∈ S F (R)
and we denote by (2.12) H(S F (R))

the set of all such Hermitian operators.

With these preparations, we can now proceed to obtain the Heisenberg uncertainty relation for arbitrary operators A, B ∈ H(S F (R)).

Given A ∈ L(S F (R)) and ψ ∈ S F (R), we denote

(2.13) < A > ψ = < ψ, Aψ > ∈ C F
and call it the expectation value of A in the state ψ. Further, we denote

(2.14) ∆ ψ A = ( < A 2 > ψ -(< A > ψ ) 2 ) 1/2
and call it the uncertainty of A in the state ψ.

Theorem 2.1. (Extended Heisenberg Uncertainty Relation)

Given A, B ∈ H(S F (R)) and ψ ∈ S F (R) such that < ψ, ψ > = 1, then we have

(2.15) ∆ ψ A ∆ ψ B ≥ | < [A, B] > ψ |/2
where [A, B] = AB -BA.

Proof.

We start with

Lemma 2.1.

Let A ∈ H(S F (R)), ψ ∈ S F (R), then (2.16) < A > ψ ∈ R F Proof.
We have in view of (2.13), (2.11)

< A > ψ = < ψ, Aψ > = < ψ, Aψ > thus (2.4), (2.3) imply < ψ, Aψ > = < ψ, Aψ >
hence (1.13) completes the proof.

Let us now denote (2.17)

A 1 = A -< A > ψ I, B 1 = B -< B > ψ I
where I ∈ H(S F (R)) is the identity operator. Then

(2.18) A 1 , B 1 ∈ H(S F (R))
Indeed, let η, χ ∈ S F (R), then in view of (2.11) and Lemma 2.1., we have

< A 1 η, χ > = < Aη, χ > -< A > ψ < η, χ > = = < η, Aχ > -< A > ψ < η, χ > = < η, Aχ > -< η, < A > ψ χ > = = < η, A 1 χ >
and similarly with B 1 .

Next we prove

(2.19) [A 1 , B 1 ] = [A, B]
which is obtained as follows. We have from (2.17)

A 1 B 1 = AB -< A > ψ B -< B > ψ A + < A > ψ < B > ψ I thus B 1 A 1 = BA -< B > ψ A -< A > ψ B + < B > ψ < A > ψ I hence (2.19).
Further, for ψ ∈ S F (R) with < ψ, ψ > = 1, we have

(2.20) < A 1 ψ, A 1 ψ > = (∆ ψ A) 2
Indeed, in view of (2.18), we obtain

< A 1 ψ, A 1 ψ > = < ψ, (A 1 ) 2 ψ > = < ψ, (A -< A > ψ I) 2 ψ > = = < ψ, (A 2 -2 < A > ψ A + (< A > ψ ) 2 I) ψ > = = < ψ, A 2 ψ > -2 < A > ψ < ψ, Aψ > + (< A > ψ ) 2 < ψ, ψ > = = < A 2 > ψ -2(< A > ψ ) 2 + (< A > ψ ) 2 = < A 2 > ψ -(< A > ψ ) 2
In view of the above, we have for ψ ∈ S F (R) with < ψ, ψ > = 1, the relations

< ψ, [A, B]ψ > = < ψ, [A 1 , B 1 ]ψ > = < ψ, A 1 B 1 ψ > -< ψ, B 1 A 1 ψ > = = < A 1 ψ, B 1 ψ > -< B 1 ψ, A 1 ψ > = < A 1 ψ, B 1 ψ > -< A 1 ψ, B 1 ψ > = = 2i Im < A 1 ψ, B 1 ψ > Consequently | < ψ, [A, B]ψ > | = 2| Im < A 1 ψ, B 1 ψ > |
However, in view of (1.8), (1.9), (1.14), we have for z ∈ C F the relations

| Re z |, | Im z | ≤ | z |

The Extended Wintner Theorem on Unbounded Operators

In the context of the Heisenberg uncertainty relation, the non-commutativity of operators involved is crucial, since for commutative operators the respective inequality is obviously trivially satisfied, thus there is never any uncertainty, see (2.15).

Within the usual mathematical context of Quantum Mechanics given by complex Hilbert spaces H and linear operators on them, it turns out that certain simple non-commutativity relations for linear operators, such as (3.1) below, necessarily imply their unboundedness. Thus the need to consider unbounded, however, densely defined and closed operators on such Hilbert spaces, and the fundamental operators of position and momentum are well known to be among them.

Needless to say, this fact is a rather inconvenient one, since it complicates considerably the mathematical apparatus involved in Quantum Mechanics. And it was precisely the avoidance of such a complication which led von Neumann to his second mathematical model for Quantum Mechanics, namely, the one based on starting with algebras of observables, and then defining the states. Such an approach is obviosuly a reversal of the way in von Neumann's first model based on Hilbert spaces of states, where the observables are then defined as Hermitian operators.

The classical result regarding the inevitability of unbounded operators in von Neumann's first model of Quantum Mechanics is given in

Wintner's Theorem

Let H be a complex Hilbert space and A, B two bounded linear operators on it. Then there cannot be any nonzero constant c ∈ C, such that the non-commutation relation holds

(3.1) [A, B] = AB -BA = cI
where I is the identity operator on H.

The special relevance of this result is in the fact that the position and momentum operators do satisfy a non-commutation relation of type (3.1), this therefore being the reason they cannot be given by bounded operators.

Here, we give an extended version of Wintner's Theorem to the case of linear operators in L(S F (R)), see (2.2), (2.10), which are defined based on scalars given by reduced power algebras R F or C F .

The following simple linear functional analytic notions, extended to the case of reduced power algebras, will be needed.

For any wave function

ψ = 1≤h≤m γ h H h ∈ S F (R), see (2.1), (2.
2), we define its norm by

(3.2) || ψ || = sup 1≤h≤m | γ h | ∈ R + F
We note that, with values in R + F , and not merely in R + = [0, ∞), as is the usual case with wave functions ψ ∈ L 2 (R), this norm (3.2) is always well defined, regardless of the γ h being infinitesimal, finite or infinitely large elements in C F .

Now, a linear operator

A ∈ L(S F (R)) is called bounded, if and only if there exists M ∈ R + F , such that (3.3) || Aψ || ≤ M || ψ ||, ψ ∈ S F (R)
and in this case we denote by

(3.4) M A ⊆ R + F
the set of all such M .

We will also need the following partial order relation on

R F . Given u = (u n ) n∈N + I F , v = (v n ) n∈N + I F ∈ R F , we define (3.5) u ≤ v ⇐⇒ v -u ∈ R + F
Clearly, with the partial order (3.5), M A has the property

(3.6) M ∈ R + F , M ≥ M ∈ M A =⇒ M ∈ M A
The interest in dealing with M A is that, in this way, we can avoid the issue of considering the existence, and of the properties of inf M A in the reduced power algebra R F . Here we note that in the usual case of operators on Hilbert spaces, instead of (3.4) one obviously has

M A ⊆ R + = [0, ∞), thus inf M A ∈ R + always
exists, and it is denoted by ||A||, hence the above issue simply does not arise. However, as the following two easy to prove Lemmas show it, we can to a good extent avoid that issue even in the general case of arbitrary reduced power algebras R F .

Lemma 3.1.

Let ψ, ψ ∈ S F (R) and c ∈ C F , then 1) || ψ + ψ || ≤ || ψ || + || ψ || 2) || c ψ || = | c | || ψ || 3) || ψ || = 0 ⇐⇒ ψ = 0 ∈ S F (R) Lemma 3.2.
Let be any bounded operators A, B ∈ L(S F (R)) and c ∈ C F . Then the following hold

1) ∀ K ∈ M A , L ∈ M B : ∃ M ∈ M A+B : M ≤ K + L 2) ∀ K ∈ M A : ∃ M ∈ M cA : M ≤ |c|K 3) ∀ K ∈ M A , L ∈ M B : ∃ M ∈ M AB : M ≤ KL
The extension of the Wintner Theorem can now be formulated as follows Theorem 3.1.

Let A, B ∈ L(S F (R)) two bounded linear operators. Then there cannot be any nonzero constant c ∈ C F , such that the non-commutation relation holds

(3.7) [A, B] = AB -BA = cI
where I is the identity operator on S F (R).

Proof.

Obviously, it suffices to consider the case c = 1. Let us then assume A, B ∈ L(S F (R)) two bounded linear operators, such that

(3.8) AB -BA = I
Then by induction, one obtains

(3.9) nB n-1 = AB n -B n A, n ≥ 1
Indeed, for n = 1, the relation (3.9) reduces to (3.8). Assuming now that (3.9) holds for a certain n ≥ 1, we have then

(3.10) (n + 1)B n = nB n-1 B + B n I = = (AB n -B n A)B + B n (AB -BA) + AB n+1 -B n+1 A
Now in view of Lemma 3.2. applied to (3.9), one obtains

(3.11) ∀ n ≥ 1, K ∈ M A , L ∈ M B : ∃ M ∈ M B n-1 : nM ≤ 2KLM
consequently, we must have M = 0, for some n ≥ 1. And then, (3.9) implies B n-1 = 0, and thus successively, B = 0, and finally I = 0, which of course is absurd.

Question on Two Fundamental Physical Constants

Re-formulating Physics in terms of scalars given by reduced power algebras leads naturally to the following two questions

• Is it possible that Planck's constant h is in fact an infinitesimal in some reduced power algebra R F ?

• Is it possible that the maximum speed of propagation of physical effects is not finite, but rather an infinitely large quantity in some reduced power algebra R F ?

The motivation for these two questions appears quite natural, as soon as one becomes more familiar with the non-Archimedean structure of reduced power algebras, [3][4][5][6][7][8][9][10]. Indeed, that non-Archimedean structure leads to the presence of three types of elements in such algebras, namely : infinitesimals, finite elements, and infinitely large elements.

The essential fact in this regard, however, is that the above classification in three types of elements is relative. Namely, it is implied by the fact that, when constructing reduced power algebras R F , one starts by defining the usual real numbers in R as being the finite ones. Indeed, such reduced power algebras have a highly complex and rich self-similar structure. And it is easy to see that, due to that structure, one is in fact not obliged to choose the usual real numbers in R as being the finite ones. On the contrary, that self-similar structure renders the concept of "finite elements" highly relative, by allowing a wide range of other choices. In this way, elements which in a choice are finite, may become infinitesimal or infinitely large in other choices, and vice-versa.

Consequently, when using reduced power algebras in Physics, thus non-Archimedean scalar structures, one is no longer obliged to have both the Planck constant and that of the maximum speed of propagation of physical effects finite, and thus having the only possible difference between between them reduced to a large but finite factor.

Question on the Status of the Heisenberg Uncertainty in Quantum Mechanics

As shown in [2], it is possible to present most of Quantum Mechanics without recourse to the Heisenberg uncertainty principle. This fact seems particularly surprising to most of physicists, and so far, it seems not to have been given a proper consideration.

Part III : No-Cloning in Reduced Power Algebras

Preliminaries

The No-Cloning property in Quantum Computation is known not to depend on the unitarity of the operators involved, but only on their linearity. Based on that fact, here it is shown that the No-Cloning property remains valid when Quantum Mechanics is re-formulated within far wider frameworks of scalars, namely, one or the other of the infinitely many reduced power algebras which can replace the usual real numbers R, or complex numbers C.

A remarkable feature of the so called No-Cloning property in Quantum Computation, [3,2], is that it is but a rather elementary and direct consequence of the linearity property of unitary operators on finite dimensional complex Hilbert spaces, and in fact, it does not require that the respective operators by unitary. The fact that unitary operators are involved in Quantum Computation is natural and unavoidable, since in Quantum Mechanics it is axiomatic that the evolution of a quantum systems which is not under measurement is given by such operators, being described by the Schrödinger equation.

Based on the above elementary fact underlying usual No-Cloning, here we shall show that the No-Cloning property remains valid when Quantum Mechanics is re-formulated within what appears to be a far more wide and appropriate framework of scalars, namely, any one of the infinitely many algebras which belong to the class of reduced power algebras, [4][5][6][7][8][9][10][11][12].

One of the essential features of scalars in algebras of reduced powers is that, in addition to being finite, just as the usual real or complex numbers, such scalars in algebras of reduced powers can also be infinitesimal, or on the contrary, infinitely large. Consequently, vast opportunities for algebraic operations are opened, and also, for appropriate physical interpretations. For instance, one may consider the possibility that the Planck constant h is a nonzero positive infinitesimal, and/or the speed of light c is positive and infinitely large, [12, section 4].

An Extension of the No-Cloning Property

We recall that the field R of usual real numbers can be extended into any of the infinitely many possible so called reduced power algebras R F , where F suitable filters on the set N of natural numbers, see Appendix, and for further details [12, pp. 3-6], [4][5][6][7][8][9][10][11]. Similarly, the field C of usual complex numbers can be extended into any of the infinitely many possible reduced power algebras C F . Furthermore, some of these algebras R F and C F are themselves fields, namely, when F are ultrafilters on the set N of natural numbers.

Let us now recall that in usual Quantum Computation, a quantum register of one qubit is represented as a vector in the complex Hilbert space C 2 . And in general, a quantum register of n ≥ 1 qubits is represented by the n-fold tensor product (2.1)

H n = C 2 ⊗ . . . ⊗ C 2 ≈ C 2 n
Here, we shall replace such usual quantum registers of n ≥ 1 qubits by the larger spaces

(2.2) H F , n = (C F ) 2 ⊗ . . . ⊗ (C F ) 2
with n ≥ 1 factors, which again are vector spaces over C, as can easily be seen in [12, pp. 3-6], [4][5][6][7][8][9][10][11]. Furthermore, they possess an extended scalar product (A.20) -(A.26) which gives them properties similar with the usual Hilbert spaces, properties sufficient in order to establish the extended version of the No-Cloning property.

What is important to note is that, since the No-Cloning property does not in fact require the unitarity of the operators involved, but only their linearity, we can proceed with the extension of the No-Cloning property to quantum registers given by the vector spaces H F , n in (2.2), without having to consider on them any usual Hilbert space structure, and instead, by only using the above mentioned extended Hilbert space structure of these spaces.

In order to make clear this argument, let us briefly recall the usual No-Cloning property, [3].

First, let us note that, scientists are on occasion giving names to new phenomena in ways which are not thoroughly well considered, and thus may lend themselves to misinterpretation. One such case is, unfortunately, with the term No-Cloning used in Quantum Computation. What is in fact going on here is that, quite surprisingly, quantum computers do not allow the copying of arbitrary qubits. And here by "copying" one means the precise reproduction any finite number of times of a given arbitrary qubit, a reproduction which does not destroy the original qubit which is being reproduced. Thus a more proper term would be the somewhat longer one of no arbitrary copying. Yet in spite of that, plenty of copying can be done by quantum computers, as will be seen in the sequel.

In order better to understand the issue, let us start by considering copying classical bits. For that purpose we can use the classical ver-sion of the quantum CNOT gate, [2,3], operating this time on bits a, b ∈ { 0, 1 }, namely Now, if we fix b = 0, then for an arbitrary input bit a ∈ { 0, 1 }, we shall obtain as output two copies of a.

Strangely enough, a similar copying of arbitrary quantum bits cannot be performed by quantum systems, as was discovered in 1982 by W K Wooters and W H Zurek, [2,3]. Of course, as well known, [2,3], each qubit contains a double infinity of classical information since it can be an arbitrary point on the Bloch sphere, which is much unlike the situation with one single bit. In this way, the ability to copy arbitrary qubits is considerably more demanding than copying arbitrary classical bits.

Let us now turn to this issue in some more detail. First we present a simple and somewhat intuitive argument. We assume that we have a quantum system S which allows one qubit at input and has one qubit at output. The output facility we shall use as a "blank sheet" on which we want to copy an arbitrary input qubit | ψ > ∈ C 2 . We can assume that the initial state of the "blank sheet" at the output is given by a fixed qubit | χ 0 > ∈ C 2 . Thus we start with the setup

| ψ > | χ 0 > S Fig. 2.2.
and would like to end up with the setup

| ψ > | ψ > S Fig. 2.3.
However, as quantum processes evolve through unitary operators when not subjected to measurement, it means that we are looking for such a unitary operator U : C 2 ⊗ C 2 -→ C 2 ⊗ C 2 , and one which would act according to

(2.3) U ( | ψ > ⊗ | χ 0 > ) = | ψ > ⊗ | ψ >, | ψ > ∈ C 2
Before going further, let us immediately remark here that a unitary operator U , which therefore is linear, is not likely to satisfy (2.3), in view of the fact that this is a nonlinear relation in | ψ > ∈ C 2 , and in particular, its left hand term is linear in | ψ >, while its right hand term is a quadratic in | ψ >.

And now, let us return to a more precise argument. Since | ψ > ∈ C 2 is assumed to be arbitrary in (2.3), we can write that relation for any

| ψ 1 >, | ψ 2 > ∈ C 2 .
Thus we obtain (2.4)

U ( | ψ 1 > ⊗ | χ 0 > ) = | ψ 1 > ⊗ | ψ 1 > U ( | ψ 2 > ⊗ | χ 0 > ) = | ψ 2 > ⊗ | ψ 2 >
Now if we take the inner product of these two relations and recall that U was supposed to be unitary, we obtain

(2.5) < ψ 1 | ψ 2 > = ( < ψ 1 | ψ 2 > ) 2
which implies that either < ψ 1 | ψ 2 > = 0, or < ψ 1 | ψ 2 > = 1. This means that the two arbitrary quantum states | ψ 1 >, | ψ 2 > ∈ C 2 are always either orthogonal, or identical from quantum point of view, which is clearly absurd.

The general and rigorous argument is as follows. We consider a quantum system whose state space is C m , for a certain integer m ≥ 2. Further, we fix in this state space an arbitrary orthonormal basis | ψ 1 >, . . . , | ψ m > ∈ C m . Finally, we assume that the state | ψ 1 > will function as the "blank sheet" on which we want to copy arbitrary states | ψ > ∈ C m .

Then the desired copying machine of arbitrary states in C m will be given by a unitary operator U : C m ⊗ C m -→ C m ⊗ C m , for which we have

(2.6) U ( | ψ > ⊗ | ψ 1 > ) = | ψ > ⊗ | ψ >, | ψ > ∈ C m
And now we can prove that for n ≥ 2, there does not exist such a copying machine U . Indeed, if we assume that n ≥ 2, then we do have at least the two orthonormal states | ψ 1 >, | ψ 2 > ∈ C m . Thus (2.6) gives (2.7)

U ( | ψ 1 > ⊗ | ψ 1 > ) = | ψ 1 > ⊗ | ψ 1 > U ( | ψ 2 > ⊗ | ψ 1 > ) = | ψ 2 > ⊗ | ψ 2 > U ( ( | ψ 1 > + | ψ 2 > ) ⊗ | ψ 1 > ) = = ( | ψ 1 > + | ψ 2 > ) ⊗ ( | ψ 1 > + | ψ 2 > )
Now the linearity of U gives together with the first two relations above (2.8)

U ( ( | ψ 1 > + | ψ 2 > ) ⊗ | ψ 1 > ) = = U ( | ψ 1 > ⊗ | ψ 1 > ) + U ( | ψ 2 > ⊗ | ψ 1 > ) = = | ψ 1 > ⊗ | ψ 1 > + | ψ 2 > ⊗ | ψ 2 >
Thus (2.8) with the last relation in (2.7) imply (2.9)

( | ψ 1 > + | ψ 2 > ) ⊗ ( | ψ 1 > + | ψ 2 > ) = = | ψ 1 > ⊗ | ψ 1 > + | ψ 2 > ⊗ | ψ 2 >
or in other words First, in the more general second proof, we did not use the fact that U is unitary, and only made use of its linearity, when we obtained (2.8). In the first proof, on the other hand, the fact that U is unitary was essential in order to obtain (2.5).

Second, it is important to understand properly the meaning of the above limitation implied by No-Cloning. Indeed, while it clearly does not allow the copying of arbitrary qubits, it does nevertheless allow the copying of a large range of qubits. given by c (i, 1) = (i, i), with 1 ≤ i ≤ n. Then clearly, c is injective on the domain on which it is defined. Therefore, c can be extended to the whole of I × I, so as still to remain injective, and in fact, to become bijective as well. And obviously, there are many such extensions when n ≥ 2. Now we can define a mapping U by

U ( | ψ i > ⊗ | ψ j > ) = | ψ k > ⊗ | ψ l >
where 1 ≤ i, j ≤ n and c (i, j) = (k, l). Since c is bijective on I × I, this mapping U will be a permutation of the respective basis in C m ⊗ C m , therefore it extends in a unique manner to a linear and unitary mapping

U : C m ⊗ C m -→ C m ⊗ C m
And now it follows that

U ( | ψ i > ⊗ | ψ 1 > ) = | ψ i > ⊗ | ψ i >, 1 ≤ i ≤ n
thus indeed U is a copying machine with the "blank sheet" | ψ 1 >, and it can copy onto this "blank sheet" all the qubits in the given orthonormal basis | ψ 1 >, . . . , | ψ n > of C m . And in any such basis, with the exception of the fixed "blank sheet" | ψ 1 >, all the other qubits | ψ 2 >, . . . , | ψ n > are arbitrary, within the constraint that together they have to form an orthonormal basis.

Returning now to the extended situation in (2.2), we obtain the fol-

  Fig. 2.1.

( 2 . 10 )

 210 | ψ 1 > ⊗ | ψ 2 > + | ψ 2 > ⊗ | ψ 1 > = 0which is obviously false.Let us point out two facts with respect to the above no-cloning result.

  For instance, in terms of the second proof, let I = { 1, . . . , n } be the set of indices of the respective orthonormal basis| ψ 1 >, . . . , | ψ n > ∈ C mFurther, let us consider the partially defined function c : I × I -→ I × I

The general construction of reduced power algebras goes as follows, [2][3][4][5][6][7][8][9][10][11]. Let Λ be any infinite set. Let F be any filter on Λ, such that Then, through the usual quotient construction, we obtain the reduced power algebra (A.1.4) A F = R Λ / ≈ F which has the following two properties.

The mapping

Further, on A F we have the partial order which is compatible with the 12 algebra structure, namely

As is well known (A.1.7) A F is a field ⇐⇒ F is an ultrafilter on Λ consequently (A.1.8) A F has zero divisors ⇐⇒ F is not an ultrafilter on Λ It will be useful to consider the non-negative elements in A F , given by (A.1.9)

. Zero Divisors and Units in A F

Let F be a filter on Λ which satisfies (A.1.1) and is not an ultrafilter on Λ. Given any x ∈ R Λ , we denote

and obviously, we have the following four alternatives

Let us now consider (A.2.3.3) and define y ∈ R Λ by (A.2.5)

In the case of (A.

It follows that the set of units, or invertible elements in A F is given by (A.2.9)

while the set of zero divisors in A F is given by (A.2.10)

and clearly, we have the following partition in three disjoint subsets

. Infinitesimals and Infinitely Large Scalars

The reduced power algebras A F contain strictly as a subfield the field R of usual real numbers. In addition, the reduced power algebras A F contain vast amounts of infinitesimal, as well as infinitely large scalars.

In case in (A. 1.3), and in the sequel, we replace R with C, and thus C Λ takes the place of R Λ , then we obtain reduced power algebras which contain strictly the field C of usual complex numbers. And again, the reduced power algebras will contain vast amounts of infinitesimal, as well as infinitely large scalars.

A.4. Reduced Power Fields

The following properties are equivalent :

We note that the relations (2.7) -(2.10) extend easily to (2.11), (2.12).

Appendix

For convenience, we shall recall in a particular case the construction, [4][5][6][7][8][9][10][11], as reviewed in [12, pp. 3-6], of reduced power algebras. Given any filter F on N, we define (A.1)

which is a proper ideal in the algebra R N . Thus we obtain the reduced power algebra associated to F as the quotient algebra

Furthermore, this algebra which is commutative, is also a strict extension of the field R of the usual real numbers, according to the embedding of algebras

In a similar manner one can obtain reduced power algebras extending the field C of the usual complex numbers. Namely, let us denote by

which is a proper ideal in the algebra C N . Thus we obtain the reduced power algebra associated to F as the quotient algebra

Furthermore, this algebra which is commutative, is also a strict extension of the field C of the usual complex numbers, according to the embedding of algebras

We now establish a natural connection between the algebras R F and C F .

In this regard, we note the following connection between the ideals I F and J F . Namely (A.7)

where u n , v n ∈ R. It follows that we have the algebra homomorphisms (A.8)

as well as the algebra embeddings

Let us also define the surjective linear mapping (A.12)

As a consequence, we obtain

Lastly, we can define the absolute value on C F , by the mapping (A.14)