On the Present State of Fundamental Physics

The laws of physics are presently described in the framework of two main theories, namely the theory of relativity (special^1-2 and general,³ which include classical mechanics) and quantum mechanics^4-6 (developed into quantum field theories). Both edifices are extremely efficient and precise in their predictions; the constraints imposed by special relativity have even been incorporated in a relativistic quantum theory. But these two theories are founded on completely different grounds, even contradictory in apinpearance, and make use of a completely different mathematical apparatus.

General relativity is a theory based on fundamental physical principles, namely the principles of general covariance and of equivalence. Its mathematical tools come as natural achievements of these principles. On the contrary quantum mechanics is, at present, an axiomatic theory. It is founded on purely mathematical rules which, up to now, are not understood in terms of a more basic mechanism.

This leads to a strong dichotomy in physics: two apparently opposite worlds cohabit, the classical and the quantum. In particular gravitation, so clearly and accurately described by Einstein's theory of general relativity,³ has escaped up to now any admissible description in terms of the quantum field-particle approach. Conversely, our understanding of the electromagnetic, weak and strong fields has made huge progress in the framework of quantum gauge theories,^7-9 while all classical attempts to unification (e.g., of gravitation and electromagnetism) have ended in failure.

These and other signs indicate, in our opinion, that physics is still in infancy. Several great problems, maybe the most fundamental ones, are still completely open. There is at present no theory able to make predictions about the two "tails" of the physical world, namely elementarity and globality, i.e., at the smallest and largest time scales and length scales.

At small scales, the "standard model" of elementary particles, based on quantum chromodynamics and electroweakdynamics, is able to include in its framework the observed structure of elementary particles and coupling constants (i.e., charges). But it seems, up to now, unable to predict on purely theoretical grounds either the number of elementary particles, or their masses, nor the values of the fundamental couplings. This failure is certainly related to the main failure of electrodynamics (classical and quantum): the divergence of self-energy and charge at infinite energy.¹⁰ Renormalization^11-13 was only a partial solution to the problem. By replacing in calculations the theoretical infinite charges and masses by the observed ones, it allowed physicists to predict with high precision all the other physical quantities of interest. But the problem of masses and charges was left open.

At the other end, that of very large scales, even though the current cosmological theory has known great successes, one must not forget that general relativity, being a purely local theory (its fundamental tool, the metrics element, is differential), tells us nothing about the global topology of the Universe.¹⁴ This is, with the problem of sources of gravitation (why does inertia curve space-time?), one of the limiting domains where general relativity is an incomplete theory, as recognized by Einstein himself:¹⁵ an indication of this incompleteness may be its inability to include Mach's principle, except in some particular models, while observations seem to imply that it is effectively achieved by Nature (see [57] Secs. 5.11 and 7.1).

The intermediate classical world is not devoid of open fundamental problems. Recent years have known an impressive burst in the study of dynamical chaos.^16-18 Chaos is defined as a high sensibility on initial conditions which leads to rapid divergence (e.g. exponential) of initially close trajectories, then to a complete loss of predictability on large time scales. Chaos is encountered in equations which look quite deterministic, in a large number of different domains like chemistry, fluid mechanics and turbulence, economics, population dynamics, celestial mechanics, meteorology... The challenge of chaos is that structures are very often observed in domains where chaos has developed, while ordinary methods fail to make prediction because of the presence of chaos itself. The understanding of how "order" (better: "organization") emerges from chaos is the key for the foundation of a future (still not existing) science of classical complexity. This fundamental problem is addressed in [57] Secs. 3.2, 5.6, 5.7 and 7.2.

We shall attempt to convince the reader that these questions, in the quantum, cosmological and classical complexity domains, may actually be of a similar nature. They all turn around the problem of scales, and may be traced back to a still unanswered very fundamental question: what determines the fundamental scales in Nature? A theory of scale is needed in physics. We shall propose here that Einstein's principle of relativity applies not only to laws of motion but also to laws of scale, and thus can be used as a basic stone for founding such a theory. But let us first briefly describe the present status of the theories of relativity and of quantum mechanics.

The theory of relativity

Galilean relativity, Einstein's special and general theories of relativity are successive attempts to make possible the expression of physical laws in more and more general coordinate systems. Let us recall Einstein's statement of the principle of general relativity:³ "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion". Remark that, under this form, this principle is, strictly speaking, a principle of the relativity of motion. This principle is particularly remarkable by the combination of its simplicity and its extraordinary power for deriving the most fundamental constraints which govern the physical world. Take Galileo's statement of the principle: "motion is like nothing". At first sight it may look like a trivial statement. But the explicit expression of this principle actually imposes strong universal constraints about the possible forms that physical laws can take. Since motion cannot be detected by purely local experiments, only some particular laws of transformation between inertial systems are admissible. This leads to the classical laws of Galilean physics and, adding the postulate of the invariance of some velocity c, to Einstein-Poincaré-Lorentz special relativity. Moreover ithas been demonstrated in [57] Sec. 6.4 that this additional postulate is not necessary for deriving the Lorentz transform: i.e., the Lorentz transform may be shown to be the general transformation which achieves the principle of special relativity in its Galilean form.

Special relativity leads to the constraint that no velocity can exceed some universal velocity c, which may subsequently be shown to be the velocity of particles of null mass, in particular that of light.¹⁹ Recall that the Minkowskian space-time is characterized by the invariant ds² = c²dt² - (dx² + dy² + dz²), under any change of inertial coordinate system. Then it was one of Mach's main contributions to the evolution of physics to insist on the relativity of all motions, not only of inertial ones. From general covariance and the principle of equivalence, Einstein constructed the theory of general relativity, whose equations are constraints on the possible curvatures of space-time. Einstein's equations R_mn -(1/2) R g_mn - L g_mn= c T_mn are the most general simplest equations which are invariant under any continuous and differentiable transformation of coordinate systems.

For a full account of the theory, we send the reader to textbooks as those by Misner, Thorne and Wheeler²² or Weinberg.⁵⁵ Let us only briefly recall here that, in these equations, the g_mn's are tensorial metric potentials which generalize the Newtonian gravitational scalar potential. The general relativistic invariant reads, in terms of Einstein's convention of summation on identical lower and upper indices

ds² = g_mn dx^m dxⁿ , (m,n = 0 to 3).

In general relativity, the curvature of space-time implies that the variation of physical beings (such as vectors or tensors) for infinitesimal coordinate variations depends also on space-time itself. This is expressed by the covariant derivative



which generalizes the partial derivatives. In this expression, the effect of space-time (i.e., of gravitation) is described by the Christoffel symbols



which play the role of the gravitational field. The covariant derivatives do not commute, so that their commutator leads to the appearance of a four-indices tensor, the so-called Riemann tensor R^l_mnr :

(D_m D_n- D_n D_m ) A_r= R^l_rnmA_l .

Contraction of the Riemann tensor yields the Ricci tensor R_mn = g^lrR_lmrn :

(1)

while the quantity R=g^mnR_mn is the scalar curvature. Einstein's equations state that the energy-momentum tensor T_mn is equal, up to the constant c = 8pG/c⁴, to the geometric Einstein tensor given in the first member of (1) , in which L is the cosmological constant. The Einstein tensor and the energy-momentum tensor are conservative in the covariant sense. Einstein's equivalence principle of gravitation and inertia is expressed by the fact that one may always find a coordinate system in which the metric is locally Minkowskian, and that in such a system the equation of motion of a free particle is that of inertial motion, Du_m = 0, where u_m is the four-velocity of the particle. Written in any coordinate system, this equation becomes the geodesics equation

d²x^m/ds² + G^m_nr (dxⁿ/ds) (dx^r/ds) = 0 .

Note that the principle of relativity, in Einstein's formulation, applies to the "laws of Nature", which Einstein carefully distinguishes from the equations of physics. Laws of Nature are assumed to exist independently from the physicist (this is the first underlying working postulate of any "philosophy of nature") while equations of physics are the mathematical expression of our own (always perfectible) attempts at reaching them. The mathematical translation of the principle of general relativity is Einstein's principle of general covariance:³ "the general laws of nature are to be expressed by equations which hold good for all systems of coordinates, that is, are covariant with respect to any substitutions whatever".

The evolution of the principle of relativity is intrinsically linked to evolution of the concept of space-time. Any interrogation about the physics involved in the transformation of reference frames runs into an interrogation about space-time. Reversely, asking questions about space-time leads one to question relativity. In Galilean relativity, space and time are assumed to be absolute and independent concepts. The special theory^1-2 renounces such views, and introduces the concept of space-time.²⁰ But the Minkowskian space-time is still absolute, while the analysis by Mach, then by Einstein,³ clearly shows the general covariance requirement to be inconsistent with the idea of a privileged space or space-time. This leads to the space-time of the general theory which depends on the material and energetic content of the universe.

The way through which space-time properties are related to matter properties is instructive (by "matter" we mean matter and energy). It consists in attributing to space-time those properties of matter which are universal. It is the universality of the Lorentz transform, which applies not only to electromagnetic waves but also to any kind of massive particle or system, that allows the introduction of the Minkowskian space-time. In general relativity, the universal property of matter pointed out by Einstein is the curvature of trajectories. This allows one to understand the crucial role played by the deviation of light rays (and more generally by all effects of gravitation on light) in the construction of the theory²¹ and in its final acceptance. The universality of the curved nature of trajectories of particles, whether massive or not, leads to attributing the property of curvature to space-time itself. Then in the curved space-time, free
particles follow geodesical lines, in agreement with the equivalence principle.

The power of this approach is appreciable when remembering that, if one completely accepts Einstein's geometrical interpretation, the concepts of forces, of potentials and of field disappear for the benefit of the mere curved space-time: from the principle of equivalence, inertia is found back locally in a freely falling reference system, i.e., one which follows a geodesic of the Riemannian space-time. Space-time, as described by the metric potentials, may be considered as a new mathematical tool, even more profound than that of field; from it the notions of force and of potential may be finally recovered, but as approximations. If we push to its logical ends the argument of the vanishing of the field, we get Einstein's radical interpretation of the nature of gravitation, as being nothing but the manifestation of a universal property of the world, the space-time curvature.²²

Quantum physics

At present, quantum mechanics is being based on a completely different approach. Let us briefly recall (and comment) the axioms of the nonrelativistic theory (see e.g. Refs. 23-25):

(i) A physical system is defined by a state function f. Its coordinate realization, the complex wave function y (q,s,t), is often used in the non- relativistic theory. It depends on all classical degrees of freedom, q and t, and on additional purely quantum mechanical degrees of freedom s, such as spin. The probability for the system to have values (q, s) at time t is given by P = |y (q,s,t)|², so one may write y in the form
y = P^1/2e^iq. Following Feynman, one often rather uses the probability amplitude y(a,b) between two space-time events a and b. The extraordinary fact about quantum mechanics is that the full complex probability amplitude, with modulus P^1/2 and phase q, is necessary to make correct predictions, while only the square P of the modulus is observed. When an event can occur in two alternative ways the probability amplitude is the sum of the probability amplitudes for each way considered separately:

y = y₁ + y₂ => P = P₁ + P₂ + y₁ y₂ + y₂ y₁

The two new rectangular terms additional to the classical terms P₁ + P₂ are at the origin of interferences and more generally of quantum coherence. Conversely, when it is known whether one or the other alternative is actually taken, the composition of probabilities takes the classical form P = P₁ + P₂ .

(ii) If y₁ and y₂ are possible states of a system, then y = ay₁ +by₂ is also a state of the system.

(iii) Physical observables are represented by linear Hermitian operators, W, acting on the state function. For example there corresponds to momentum p_i the complex operator . This is another manifestation of the mysterious character of quantum rules that, to a real momentum, there corresponds a complex operator acting on the complex probability amplitude.

(iv) Results of measurements of physical observables are given by any of the eigenvalues of the corresponding operator, W y = w_i y.

(v) Any state function can be expanded as y = S_n a_ny_n in an orthonormal basis, and a_n² records the probability that the system is in the n^th eigenstate.

(vi) The time evolution of the system satisfies Schrödinger's equation
where the Hamiltonian H is a linear Hermitian operator.

(vii) Immediately after a measurement, the system is in the state given by the first measurement. This seventh axiom ("Von Neumann's axiom") is forgotten in many text books though it is necessary to account for experiments: for example after a spin measurement, the spin remains in the state given by the measurement; just after a measurement of position (at t+δt, δt → 0), a particle is in the position given by the measurement. Its absence may give a false impression of quantum mechanics as a theory where precise predictions can never be done, while this depends on the pure or mixed character of the state of the system. It underlies the phenomenon of reduction of the wave packet.²⁶

These axioms have well-known philosophical consequences (or better, they are a self-consistent mathematical transcription of what experiments have told us about the microphysical world: the philosophical consequences originate, in the present thinking frame, from observations). Two realizations of the state function, the coordinate and momentum representations, are particularly relevant in this respect. The position and momentum wave-functions may be derived one from the other by reciprocal Fourier transform. From this comes the Heisenberg inequality

s_x s_p >= h / 2

which implies the non-deterministic character of quantum trajectories. Note also that the solution of Schrödinger's equation for a free particle leads to the introduction of the de Broglie length and time: The phase of the wave-function writes q = (p x - E t) / h, where p and E are the classical momentum and energy of the particle. The de Broglie periods, h/p and h/E, correspond to a phase variation of 2p. Throughout this book, we shall call "de Broglie length and time" the quantities

l = h / p ; t = h/ E

such that the quantum phase for a free particle writes q = (x /l- t/t). (Recall that historically, the Schrödinger equation was constructed as the equation whose de Broglie wave, obtained earlier, was a solution). The de Broglie scale may be generalized to more complicated systems: it can be identified with the characteristic transition scale occurring in the quantum phase.

It is clear from the above axioms that most of the quantum mystery may be traced back to the mere question: "where does the complex plane of quantum mechanics lie?" We shall in this book propose a solution to this puzzle by showing that a complex plane naturally emerges in space-time from the simplest prescription aimed at describing a non-differentiable space-time; this allows one to obtain Schrödinger's equation as the form taken by the fundamental equation of dynamics written in such a non-differentiable frame (see [57] Chapter 5).

Quantum mechanics is an axiomatic theory rather than a "theory of principle" as relativity. We do not understand the physical origin of these axioms: we only know that they work, i.e., that the theory developed from them has a high predictive power and is remarkably precise. The mathematical beings of the present quantum theory are defined in an abstract space of state, so that the part played by the standard space-time has apparently been deeply decreased. Quantum physics in its present form describes rather the intrinsic properties of microphysical objects embedded in a space-time which is a priori assumed to be Euclidean or Minkowskian. Though it has already been suggested that the structure of the microphysics space-time could be foamy or gruyere cheese-like^27-29, this was assumed to hold only at the level of Planck's length and time. These ideas have now been developed into attempts to build a theory of quantum gravity (see e.g. Ref. 30 and references therein), whose preferential domain of application would be the very early universe.

Geometry and Microphysics

In spite of the advantages one may find in a space-time theory, the several unsuccessful attempts to unify gravitation and electromagnetism from a geometrical approach based on curvature and/or torsion (see e.g. Ref. 31) finally convinced physicists that such an approach has to be given up. The parallel success of quantum gauge theories led to the hope that unification may rather be reached only from the quantized field-particles approach, and that gravitation itself should be quantized in the end. However, among the various causes of the failure of previous geometrical attempts, two may be pointed out in the light of the hereabove remarks:

(i) the observed properties of the quantum world cannot be reproduced
by Riemannian geometry;

(ii) the space-time approach cannot be based on particular fields, but on those properties of matter which are universal. It is thus clear that any new insights about the nature of the microphysical space-time may only be gained provided new concepts are introduced.

In this book we shall review the principles and the first results of a new attempt at reconsidering the conclusion that a geometrical approach of the quantum properties of microphysics is impossible. We suggest possible ways towards the construction of a spatio-temporal theory of the microphysical world, basing ourselves on the concept of fractal space-time in connection with the suggestion of an extension of the principle of relativity. The crucial new ingredient of our approach with respect to present standard physics (classical and quantum) is that we assume that space-time is non-differentiable. Moreover it has been demonstrated ([57] Sec. 3.10) that non-differentiability implies an explicit dependence of space-time on scale.

In this quest, our main lead will be Einstein's principle of relativity. But we shall take here relativity as a general method of thinking, rather than as a particular theory. In a relativistic approach to physics, one tries to analyze what, in the expression of physical laws, depends on the particular reference system used, and which properties are independent of it. We shall show that the principle of relativity applies not only to motion, but also to scale transformations, once the resolution of measurements is defined as a state of the coordinate system.
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Quoted from: Nottale L., 1993, "Fractal Space-Time and Microphysics: Toward of Theory of Scale Relativity", Chap. 2 (© World Scientific, 1993).