aa r X i v : . [ phy s i c s . h i s t - ph ] J a n ON THE CONCEPT OF LAW IN PHYSICS Claus Kiefer
Institute for Theoretical Physics,University of Cologne,Z¨ulpicher Strasse 77, 50937 K¨oln, Germany.
Abstract
I discuss the main features of the concept of law in physics. Fun-damental laws from Newtonian mechanics to general relativity are re-viewed. I end with an outlook on the new form of laws in the emergingtheory of quantum gravity. To appear in the Proceedings of the conference
The Concept of Law in Science , Hei-delberg, 4–5 June 2012.
Laws of Nature
The concept of law is widespread in both the sciences and the humanities.When one talks about laws of Nature, however, one usually refers to physics.What is a physical law? Richard Feynman, in his well known book
TheCharacter of Physical Law writes ([1], p. 13): “There is also a rhythm and apattern between the phenomena of nature which is not apparent to the eye,but only to the eye of analysis; and it is these rhythms and pattern which weshall call Physical Laws.” As a prototype of a physical law, Feynman statesthe law of gravitation.As is evident from Feynman’s quote, one needs a certain degree of ab-straction from the phenomena to discern the laws of Nature. Without theeye of analysis, physical laws cannot be found.In this essay, I shall briefly summarize the status of laws of Nature inmodern physics and speculate about the development of new laws. A centralrole is there indeed played by gravitation. On the one hand, Einstein’s the-ory of general relativity has introduced a dynamical spacetime into physicsand has thus dramatically changed our attitude towards the formulation offundamental laws. On the other hand, one expects that the consistent uni-fication of general relativity with quantum theory will lead to a completelynew type of law. For this reason, I shall discuss some aspects of fundamentallaws as they appear in one approach to quantum gravity.Physical laws are formulated within a physical theory or a set of theories.A theory consists of a set of mathematical equations and a set of mappingrules to phenomena in Nature. In the ideal case, these rules include a state-ment about their domain of validity.In his famous book
Il Saggiatore , Galileo Galilei has introduced the pic-ture of Nature as a book written in mathematical language. This must,however, not be interpreted too literally. The mathematical language is notunique, and the same phenomena can be described in different ways. Agood example is gravitation. In Newtonian terms, the motion of planets isdescribed by differential equations containing an action at a distance. InEinsteinian terms, the planets move on geodesics in spacetime. If gravity iscombined with quantum theory, yet another mathematical picture emerges.There is thus not a one-to-one relation between mathematics and reality.This is clearly expressed by a famous quote from Albert Einstein, who writes([2], p. 119–120)Insofern sich die S¨atze der Mathematik auf die Wirklichkeit beziehen,sind sie nicht sicher, und insofern sie sicher sind, beziehen sie sich1icht auf die Wirklichkeit. According to Einstein, a certain degree of intuition is needed to find thecorrect laws of Nature; they cannot just be read off from the phenomena.Still, physical laws are not invented, but discovered, because they reflectproperties of the real world, not just our imagination. In contrast to this,mathematical concepts are, in my opinion, invented. Why there are lawsof Nature at all, is not obvious; nor is it a priori clear that we are able todiscover them.One can distinguish between physical laws at different levels. Here, weare mainly concerned with the fundamental laws, that is, laws that describethe fundamental interactions; examples are the laws of gravitation and elec-trodynamics. It is an open issue whether all these fundamental laws can beunified to one fundamental theory, often called ‘theory of everything’. If thishappened, it would be the ultimate triumph of the reductionist programmein physics.At a different level, one has effective laws such as the Second Law ofthermodynamics. As we shall briefly discuss below, the Second Law seemsto be a consequence of particular boundary conditions of our world, and itis open whether it can be derived in a different way from structures of a newtheory, such as quantum gravity.Yet another level concerns emergent laws for complex systems. They can,in principle, be derived from the fundamental laws, but show features thatgo much beyond those laws. In the words of Paul Anderson ([4], p. 395),. . . the whole becomes not only more than but very different fromthe sum of its parts.Here, we shall not discuss such emergent laws, but focus on the funda-mental physical laws from Newtonian mechanics to quantum gravity.
A most important feature of our physical theories is the separation of thedescription into dynamical laws and initial conditions. This was expressedvery clearly in Eugene Wigner’s Nobel speech ([3], p. 7–8), In so far the theorems of mathematics refer to reality, they are not certain, and in sofar they are certain, they do not refer to reality.
Principia ([7], p. 623),Absolute, true, and mathematical time, of itself, and from itsown nature, flows equably without relation to anything external.. . . Absolute space, in its own nature, without relation to anythingexternal, remains always similar and immovable. . . .Let us consider Newton’s second law of motion for the motion of a set of N particles described by their positions x i , i = 1 , . . . , N , m i d x i d t = F i . (1)3he force F i on the i th particle is here assumed to be given . In the importantcase of gravitational interaction, it reads F i = − G X j = i m i m j | x i − x j | x i − x j | x i − x j | . (2)Because (1) is a differential equation of second order in time, its solutionis determined if position and velocity are specified at a particular moment oftime.The notions of absolute space and absolute time were criticized at severaloccasions in the history of science, mainly because these notions involve ab-solute (non-dynamical) elements. Among the critics were Berkeley, Leibniz,and Mach. Since, however, Newton’s mechanics was extremely successful,attempts to formulate an alternative mechanics did not go very far [7]. Onlyafter the advent of general relativity did people investigate models of classicalmechanics without absolute space and time [8].Besides gravitation, the only fundamental interaction that manifests it-self at a macroscopic level is electrodynamics. It is described by the set ofMaxwell’s equations, ∇ B = 0 , ∇ × E + 1 c ∂ B ∂t = 0 , ∇ E = 4 πρ , ∇ × B − c ∂ E ∂t = 4 πc j , (3)where B and E are the magnetic and electric field, respectively. In con-trast to Newton’s equations, these are equations for local fields. AlreadyMaxwell’s contemporaries were impressed by the fact that these equationsencode all the phenomena related to electricity, magnetism, and optics. Oneof the main features of new fundamental laws is the fact that they can pre-dict the occurrence of new phenomena. In the case of Maxwell’s equations,these include the generation of radio waves, which proved to be of enormoustechnological significance.In the formulation of physical laws, symmetry principles play a key role.Otherwise, it would almost be impossible to devise the correct equationsout of the immense number of mathematical options. In classical mechan-ics, an important principle is the principle of relativity: the physical lawsare invariant with respect to the transformation from one inertial frame intoanother. Maxwell’s equations seem to violate this principle, because theycontain a distinguished speed – the speed of light c . It was this apparent It was Boltzmann who cited Goethe’s
Faust : “War es ein Gott, der diese Zeichenschrieb?”. t that occurs in the Schr¨odinger equation,ˆ H Ψ = i ~ ∂ Ψ ∂t , (4)is nothing but the time parameter of (1).The Schr¨odinger equation (4) is a deterministic equation: if the quantumstate Ψ is given at any particular instant of time, the solution follows for anyother time value, both before and after that instant. The interpretation ofΨ is, however, drastically different from classical fields such as E or B , be-cause it is defined not in spacetime, but on a high-dimensional configurationspace. Its connection with classical quantities is described by the probabilityinterpretation. The emergence of classical behaviour is given by the processof decoherence [9].If special relativity is combined with quantum theory, one arrives at quan-tum field theory. Here, four-dimensional flat Minkowski space is used as arigid classical background on which the dynamics of the quantum fields isdefined.As many authors, in particular Albert Einstein, have noted, it is notnatural to envisage something that can act but which cannot be acted upon(as is the case for Minkowski space). The situation changes dramaticallywith general relativity, to which we now turn.5 General Relativity
In general relativity, the gravitational field is described by the geometry of adynamical four-dimensional spacetime. The fundamental equations are a setof ten coupled partial differential equations for the metric g µν . In standardnotation, these Einstein field equations read R µν − g µν R + Λ g µν = 8 πGc T µν . (5)For the first time, one is confronted with equations that are not formulatedon a given spacetime, but equations that describe spacetime itself. Oneimpressive example is the existence of gravitational waves, which describethe propagation of pure curvature without matter. As John Wheeler alwaysemphasized, space tells matter how to move, and matter tells space how tocurve.In spite of the complex nature of the Einstein field equations, a well-defined initial value problem (‘Cauchy problem’) can be formulated. Themetric coefficients g µν ( x ) can be determined uniquely (up to coordinatetransformations) from appropriate initial data. An important feature in thiscontext is the presence of four (at each space point) constraints . These con-straints arise from the fact that the theory is invariant under four-dimensionaldiffeomorphisms (‘coordinate transformations’). The initial data consist ofthe three-dimensional metric, the second fundamental form, and matter de-grees of freedom on a spacelike hypersurface that satisfy the four constraints.In this way, spacetime itself is constructed from initial data. The existence ofa well-defined Cauchy problem is of special relevance for numerical relativity,which is concerned with processes such as the evolution of two black holesorbiting each other.General relativity is a very successful theory. With perhaps the excep-tion of dark matter and dark energy, it describes all known gravitationalphenomena. But it behaves also in an exemplary manner with respect toits limits. From general theorems (‘singularity theorems’), one knows thatthere are situations in which the theory breaks down [10]. These are, in fact,important situations because they apply to the origin of our Universe and tothe interior of black holes. For these and other reasons, one expects that thelaws of gravity are, at the most fundamental level, not exactly described byEinstein’s equations. One way to arrive at a more fundamental theory thangeneral relativity is to take quantum theory into account. This will be thesubject of the last section. 6 Quantum Gravity and Beyond
General relativity and quantum theory cannot both be exactly valid. Onereason is what usually is referred to as the ‘problem of time’ [11]. Time isabsolute in quantum mechanics (spacetime in quantum field theory), but itis dynamical in general relativity (as part of the dynamical spacetime). Sowhat happens in situations where both theories become relevant?If one keeps the linear structure of quantum theory and looks for a quan-tum wave equation that gives back Einstein’s equations in the semiclassicallimit, one arrives at a quantum constraint equation of the general formˆ H Ψ = 0 . (6)This equation is known as the Wheeler–DeWitt equation [11]. It has someamazing properties. The full quantum state Ψ of gravity and matter de-pends on the three-dimensional metric only, but is invariant under three-dimensional coordinate transformations. It does not contain any externaltime parameter t . The reason for this ‘timeless’ nature is obvious. In generalrelativity, a four-dimensional spacetime is the analogue to a particle trajec-tory in mechanics. After quantization, the trajectory vanishes, and so doesspacetime. What remains is space, and the configuration space is the spaceof all three-geometries [12]. A constraint equation of the form (6) also occursin loop quantum gravity [11].To give a particular example, let us formulate the Wheeler–DeWitt equa-tion for a simple cosmological model. For a closed Friedmann–Lemaˆıtre uni-verse with scale factor a and a massive scalar field φ , this equation reads afteran appropriate choice of units as follows (Λ is the cosmological constant):12 (cid:18) ~ a ∂∂a (cid:18) a ∂∂a (cid:19) − ~ a ∂ ∂φ − a + Λ a m a φ (cid:19) ψ ( a, φ ) = 0 . (7)The timeless nature is evident. The cosmological wave function only dependson the two variable a and φ . As can be seen from the kinetic term, theWheeler–DeWitt equation is of hyperbolic nature (this is also true for thegeneral case). It does provides the means to define an intrinsic time , whichis distinguished by the sign in the kinetic term. This intrinsic time, however,is no longer a time given from the outset, but is defined from the three-dimensional geometry itself. In this way, it resembles what the astronomersused to call ephemeris time [8].The Wheeler–DeWitt equation thus represents a new type of physicallaw. It describes a timeless world at the most fundamental level. The usualtime parameter of physics emerges only at an approximate level and under7ery special circumstances [11]. To quote John Wheeler from his pioneeringwork ([13], p. 253),These considerations reveal that the concepts of spacetime andtime itself are not primary but secondary ideas in the structure ofphysical theory. These concepts are valid in the classical approx-imation. However, they have neither meaning nor applicationunder circumstances when quantum-geometrodynamical effectsbecome important. . . . There is no spacetime, there is no time,there is no before, there is no after. The question what happens“next” is without meaning.In spite of its timeless nature, the Wheeler–DeWitt equation can, in prin-ciple, provide the means for an understanding of the arrow of time (theSecond Law of thermodynamics mentioned above) [14]. Let us consider aFriedmann–Lemaˆıtre universe with scale factor a ≡ exp( α ) and small per-turbations symbolically denoted by x n . The Wheeler–DeWitt equation thenassumes the formˆ H Ψ = ∂ ∂α + X n − ∂ ∂x n + V n ( α, x n ) | {z } → for α →−∞ Ψ = 0 . (8)The potentials V n have the property that they vanish when the scale factorgoes to zero (i.e., near the big bang and the big crunch). This expresses afundamental asymmetry of the Wheeler–DeWitt equation. For small scalefactor, therefore, one can have solutions that are fully unentangled amongthe degrees of freedom. But for increasing a the solution becomes entangled,and one can obtain a non-vanishing entanglement entropy upon tracing outthe perturbations. Entanglement entropy can be related to thermodynamicentropy, and this entropy then increases with increasing size of the universeand thereby defines a definite direction. In the limit where an approximatetime parameter is present, this gives rise to the usual Second Law. But ifviewed from this fundamental perspective, the expansion of the universe is apure tautology.So what is the future of physical laws? The Wheeler–DeWitt equationhas not yet been experimentally tested, but it is an equation that follows ina straightforward way from the unification of quantum theory with gravity.It describes quantum effects of gravitation, but does not encompass by itselfa unification of all interactions. A candidate for a unified theory is stringtheory. The structure of the fundamental laws in this approach is not yet8ully understood, but it seems to be different from the structures discussedabove [11].Whether a fundamental ‘theory of everything’ can be found, is open. Itmay happen that such a theory will be available in this century and that thefundamental picture in physics is ‘complete’ in the sense that all phenomenacan be derived from it, at least in principle. Or it may happen that weare stuck because experimental progress becomes slower and slower and nodecision among candidates for a fundamental theory can be made. In one wayor another, it is true what Feynman wrote already in 1964 ([1], p. 172): “Theage in which we live is the age in which we are discovering the fundamentallaws of nature, and that day will never come again. It is very exciting, it ismarvellous, but this excitement will have to go.” References [1] R. Feynman (1990)
The Character of Physical Law (The M.I.T. Press,Cambridge, Massachusetts).[2] A. Einstein (1977) Geometrie und Erfahrung. In
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Quantum Gravity , third edition (Oxford UniversityPress, Oxford).[12] C. Kiefer (2009) Does time exist in quantum gravity? Second prize essayof the Foundational Questions Institute essay contest on the nature oftime. arXiv:0909.3767v1 [gr-qc].[13] J. A. Wheeler (1968) Superspace and the nature of quantum geometro-dynamcis. In
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