Bailly F., Longo G. Mathematics and the Natural Sciences: The Physical Singularity of Life

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Invariances, Symmetries, and Symmetry Breakings 153

consideration and to any systems displaying identical symmetries.

As a co nsequence, these mathematical invariants (these physica l laws

of conservation), by extricating the considered systems’ most structurally

stable quantities, constitute an ess e ntial part of their identity, that is, that

they determine them as mathematical structures associated with the sym-

metry groups (of invariance ) and as physical systems conserving certain

quantities during the changes, which aﬀect them. We thus see that the

scope of the geodesic principle extends to specifying the essential aspects

of the physical system thus considered: what remains invariant during the

undergone changes, like a sort of s table base likely to then structure all

encountered varia bilities and modiﬁcations.

As for the role of symmetries, note that many empirical systems are

submitted to the s ame symmetry groups and that the geodesic principle

enables us to regroup them under the label of the realization of a same

general model. That is, in a way, to categorize them relative to their

fundamental invariants. One can even go further by showing how these

theoretico-conceptua l s chemes are applied to the theoretical framework of

contemporary physics, to relativistic theories (including cla ssical mechan-

ics), quantum theories, and critical theories.

In this perspective, that which fundamentally characterizes relativis-

tic theories, in the more general case of Riemannian metrics, is their in-

variance under the diﬀeomorphisms of external space-time. These are re-

ducible to the Lorentz-Po incaré invariance group

and to spec ial relativity,

for Minkowskian metrics, and to the Galileo group for the Euclidean met-

Noether’s theorem is relative to the continuous transformation of symmetries which,

operating on the Lagrangians or the Lagrangian densities, preserve their equations of

movement such as derived from Hamilton’s principle (the Euler-Lagrange equations we

have just mentioned). To each of these transformations corresponds conserved physical

quantities. Thus, for instance, to the invariance of translation in space (impossibility

of physically deﬁning an absolute origin) corresponds the conservation of the kinetic

moment; to the invariance of translation in time (impossibility of physically deﬁning an

absolute origin for tim e) corresponds the conservation of energy. From a less classical

perspective, to the invariance in a change of the origin of the phases of a wave function

corresponds the conservation of the electric charge. These relationships between invari-

ances and conservations will prove to be at the origin of the gauge theories in quantum

physics. We will often go back to Noether’s theorem.

The Lorentz group corresponds to the rotations in the Minkow ski space (of which

a component is imaginary); the extension to translations in this space generates the

Poincaré group. The Maxwell equations of electromagnetism are invariant under the

Lorentz group, this being at the origin of special relativity and of its particular kinetics

(contraction of lengths, slowing of clocks, etc.).

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154 MATHEMATICS AND THE NATURAL SCIENCES

rics of classical mechanics. The geodesic principle is directly applicable here

to characterize physical trajectories in the gravitational ﬁeld. By this, gen-

eral relativity proves to be a theory of gravitation replacing the Newtonian

theory which implied absolute space and time, while gener ating the usual

dynamic conservations (energy, momentum, . . . ). It is even possible to

proceed to a uniﬁcation of this relativistic theo ry of gravitation with the

Maxwellian theory of electromagnetism by means of an increase in dimen-

sions of the reference space followed by a compactiﬁcation of the additional

dimension which we have already mentioned in section 3.1.2, by applying

the method of Kaluza and Klein (Duﬀ, 1 994) and conserving the geodesic

principle within the new space thus deﬁned.

In what concerns quantum theories (quantum ﬁeld theories), the char-

acteristic invariances are those of gauge groups, Lie groups mainly aﬀecting

the internal spaces (ﬁbers upon bases of space-time enabling us to spec-

ify quantum numbers and the co rresponding conservations) (Ryder, 1986 ).

The geodesic pr inciple applies to Lagrangian densities,

theoretical or phe-

nomenologica l, and the gauge invariances appear as the source of the rele-

vant conservation properties. It is under the aegis of such gauge theor ies and

the determination of corresponding invariance groups tha t the uniﬁcations

between quantum interactions (electroweak uniﬁcation, strong uniﬁcation

with strong interaction) can occ ur. The extr e mely important role played

in these theories by the phenomena of spontaneous symmetry breaking will

be noted.

Some classical examples of symmetry breaking Let’s brieﬂy ex-

plain, using a simple example, what we mean by this phenomenon of spon-

taneous symmetry breaking: a marble placed in unstable equilibrium at the

peak of a mound (a conﬁguration pre senting a circular symmetry), may at

the slightest of ﬂuctuations fall into a state of stable equilibrium at any

given point of the base circumference (a conﬁguration now having lost this

symmetry of rotation). One will note that each particular drop will form

a state of symmetry that is broken relative to the initial state, but if we

perform a g reat number of tr ials, e ach direction being equiprobable, the

initial symmetry will be restored on average.

Such spontaneous symmetry breaking also occurs during phase transi-

tions (matter’s changes of states). For example, a ferromagnet presents,

beyond the critical temperature (the Curie point), a null magnetization,

The Lagrangian density is the local distribution of energy whose integral over space-

time gives the (Lagrangian) action.

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Invariances, Symmetries, and Symmetry Breakings 155

and the corresponding magnetic conﬁguration has a spherical symmetry

(magnetization is the same in all directions). When we pass below the crit-

ical temperature, there appears a magnetic moment vector (a magnetiza-

tion) favoring one dire c tion in space, the direction of this vector. Likewise,

a liquid such as water presents omnidirectional symmetry properties. On

the other hand, if the liquid is cooled down below the freezing point, crys-

tals will appear, breaking this symmetry to the beneﬁt of another, a lesser

symmetry, that of the corresponding crystalline structures.

In quantum physics, symmetry breaking is supposed to generate the

mass spectrum of the intera c ting particles, via the interactions with the

Higgs bosonic interaction ﬁeld (which has not yet been experimentally

demonstrated). And numerous physicists also attribute the birth of our uni-

verse to a highly energetic spontaneous symmetry breaking of the “quantum

vacuum,” a hypothetical initial state (“Big Bang” theory).

These br e akings of s ymmetry co nstitute also the most spectacular man-

ifestation of critical phenomena and of the corres ponding critical theories

(phase transitions). Fr om a thermodynamic point of view, the geodesic

principle here takes the form of a characterization of equilibrium by the

minimization of the system’s free energy, or the form of an expression of a

criterion fo r irreversible evolution outside of equilibrium by extremization

of the speed of entropy production (Prigogine and Stengers, 1988; Nico-

lis, 1986; Nicolis and Prigogine, 1989). Fr om the view point of statistical

physics, as we have already mentioned, it is the system’s par tition function

which presents these properties of extremality. During critical trans itio ns,

the dominant physical invaria nce is namely an inva riance by dilatation in

that the correlation lengths become inﬁnite, ena bling us to put into mo-

tion the renormalization procedure (alr e ady used in the context of quantum

ﬁeld theory to eliminate certain divergences, as mentioned in Chapter 2).

This symmetry breaking has recours e to the application of the renormal-

ization (semi-)group. But another, more abstract type of invariance then

appears, this time mainly concerning the behaviors of the critical exponents

themselves, in relationship to the existence of “universality classes,” which

discard the speciﬁcities of particular systems to only retain characteristics

that are as general as the dimensionalities of reference spaces (be it the

spaces of deﬁnition of the order parameters or the embedding spaces).

Examining these examples, we can thus se e that the recognition and

application of the geodesic principle in fact opens a problem which appears

even more essential, that of the symmetries and breakings of symmetries

a physical s ystem may present or po ssess inherently. But before moving

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156 MATHEMATICS AND THE NATURAL SCIENCES

on to those mo re general consideratio ns relating to symmetries and mathe-

matical invariances of physical systems, it appears indispensable to attempt

to characterize the gnosiological imp ortance of this geodesic pr inciple. In

this regard, we have already stressed its exceptional outr each in the formal

determination of the objectivities and its organizing power for scientiﬁc

intelligibility, but these aspects do not appear limited to purely technical

reasons and seem to extend beyond the ﬁeld of physics, appearing as an

invariant of human cognition itself. This is easily conceivable if we consider

the fact, evoked above, that a geodesic characteristic, regardless of the con-

cerned reference space, is also a critical characteristic: a trajectory or a

quantity which is minimized or maximized.

Now all possible trajectories or all poss ible measurements present

generic traits in contrast to speciﬁc or critical ones. The dis tinctions be-

tween them should be clear: generic trajectories or evolutions are the many,

indistinguishable and poss ible ones, and diﬀer from the speciﬁcity of cr itical

paths, which, by this fact, ﬁnd themselves to be selected “spontaneously,”

being immediately distinguishable (a key property of geodesics). It very

well appears that from the cognitive viewpoint, within the immensity of

genericity, only profo undly speciﬁc (in par ticula r geodesic or critical) traits

are able to operate as elements of objective determinations. These range

from the role of optimal pathways for action, from the practical identiﬁ-

cation of minimal trajectories for the body’s movement, to the topological

catastrophes associated with the bo rders, which are apparent within vi-

sual perception. This is also what must have led us not only to seek and

formulate physical principles, but als o, unco nsciously, to select geometric

frameworks of reference stemming indeed from experience, but referring at

the same time to this speciﬁc characteristic, such as classical Euclidean

geometry, or leading to a search for minimal axiomatics in the characteri-

zation of formal systems. Note that Euclidean geometr y is a “critical case ”

of Riemann’s approach: its spaces have zero curvature.

Geodesics or critical paths, both in physics and in biology, currently

make interesting appro aches to complexity based on what some do not hes-

itate to call the “lands c ape paradigm” (Sherrington, 1997 ). The la nds c apes

in question correspond to multidimensional surfaces (of energetic or other

nature) and the behaviors of the systems, which they represent, are gov-

erned by the evolution of parameters within this landscape with regard to

its “passes,” “valleys,” “pe aks,” etc., that is, in fact, its set of critical points

or zones. The geodesic principle would therefore not just play an essential

role in the formal determination of the physical objectivities; it would at

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Invariances, Symmetries, and Symmetry Breakings 157

the same time meet a very profound cognitive constraint of speciﬁc discern-

ability upon generic “backgrounds,” the discerna bility corresponding to a

critical characterization.

The geodesic principle and its consequences are thus highli ghted in the

essential role of continuous symmetries and of No e ther’s theorem in the

determination o f physical systems and of their properties. We may a t-

tempt to go further in our inve stigation of the theoretical place occupied

by symmetries by including, along side these continuous symmetr ies, the ex-

amination of the discrete symmetries which can be encountered in physics

(and of their breaking), thus highlighting the essential epistemological im-

portance acquired in this ﬁeld by these questions. Indeed, the existence

of discrete s ymmetries may constrain ce rtain physical properties just as

strongly. It could be an issue of abstract symmetries: for e xample, the

symmetry or anti-symmetry pr operties under permutation of the elements

presented by the wave functions of assemblages of quanta govern their col-

lective behaviors, that is, the nature of the statistics to which they obe y:

Fermi–Dirac statistics and the Pauli fermion exclusio n principle (principles

of impe netrability and of spatial extensionality of “matter”) for antisymme-

try, Bose–Einstein statistics and Bose condensation of bosons (properties

of superposition of ﬁelds of interaction) for symmetry. But it may also

be a questio n of symmetries touching upon the prop e rties of the matter-

space-time complex itself (Cohen-Tannoudji and Spiro, 1986). Such is also

the case with the symmetries under the op e rators of parity (P) (inversion

of spatial directions), of time reversal (T), or of charge conjugation (C)

(passage from matter to antimatter). The CPT theorem spectacularly as-

sociates these three operations: it states the genera l invariance of the laws

of physics under the composition of the three operators, c urrently experi-

mentally veriﬁed in a very precise way. We know that symmetry breaking

associated with these operations involve questions as profound as those

concerning the intrinsic chirality (right-left orientation) of reference sys-

tems (breaking of P within weak interaction – dis integration for instance),

the prevalence of matter over antimatter in our universe (Sakharov, 1982),

or even the intrinsic time irreversibility of the phenomena (breaking of CP,

highlighted by the disintegration of a K mes on into two pions, for instance).

Let’s emphasize the fact that, despite the quite surpris ing aspect of such

a result and despite the important eventually derivable c osmolog ical con-

sequences, this possibility of the vio lation of CP alr e ady exists within the

framework of the contemporary so-called “standard” theory, at the level of

the distribution o f quarks into distinct families (there needs to be at least

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158 MATHEMATICS AND THE NATURAL SCIENCES

three fa milies for this possibility to a rise and the current standard model

has exactly three).

Thus, b e they continuous or discrete, symmetries take an important

place within the analysis and interpretation of contemporary physics. And,

by Noe ther’s theorem, they are related to conservation properties and, thus,

to geodesics. It is therefore legitimate to proceed to an even more attentive

examination not only of their theoretical role, but also of their cognitive

and even foundational roles, along Weyl’s views, quoted in Chapter 1.

4.2 On the Role of Symmetries and of Their Breakings:

From Description to Determination

During the preceding discussion, we have s e e n the essential role, beyond

the very expression of the geodesic principle, played by the properties of

mathematical s ymmetries a nd their breaking for the characterization of the

most fundamental of physical systems. In fac t, it very well appears that, in

the analysis of their consideration as well as in the determining role they

are called to play, we have progressively gone from the description of the

essential role which symmetries technically play within physical theories to

an attempt to interpret the way in which they organize the knowledge we

have of reality, or even the way in which they determine this physical reality

itself by contributing to constitute it. It is from this standp oint that we will

attempt to continue our investigatio n per taining to the most fundamental

objective determinations regarding the natura l s c ience s. To do so, we will

successively examine the relationships that symmetries and their breaking

maintain with profound logical structures, as well as to ex amine a possible

analysis of their role in the satisfaction of certain conditions o f possibility

for objective knowledge and thus of their role in the determination of scien-

tiﬁc re ality. In s hort, symmetries organize knowledge and its friction over

“reality”; they actually organize our world of intelligible action.

4.2.1 Symmetries, symmetry breaking, and logic

4.2.1.1 Symmetries, group struct ure, and logical equivalence

relations

If we may wholeheartedly admit the existence of relatio nships between the

structures of perc e ptio n or even of intellectual cognition and the properties

of symmetry or symmetry breaking aﬀecting the objects under examina-

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Invariances, Symmetries, and Symmetry Breakings 159

tion, there arises the opposite question of which relationships may these

properties maintain with logic stricto sensu. What we claim here is not

that logic contributes to structuring these cognitive capacities in a more

or less coherent and a priori way, but rather that it co nstitutes itself as a

formal discipline correlated to the constitution of mathematical concepts,

in their manifolded groundings. And this since ancient Greece .

We know that the mathematical gr oup structure constitutes the most

adequate means for the study and treatment of s ymmetries and that it plays

an essential role in the characterizatio n of the invariances of a system, to

an extent where some authors have made it into a fundamental element of

all scientiﬁc knowledge (Ullmo, 1969). Let’s brieﬂy demonstrate how this

mathematical structure is itself in relationship with a more strictly logical

structure, the equivalence relation. Let’s recall that a relation R (be tween

two terms x and y) is said to be an equivalence relation if it is:

i) Reﬂexive (xRx);

ii) Symmetrica l (xRy) implies (yRx);

iii) Transitive (xRy) and (yRz) imply (xRz).

Moreover, it is demonstrated that an equivalence rela tion presents the prop-

erty of partitioning the set o f elements to which it is applied into disjointed

subsets of which the reunion reconstitutes the whole set. Now, with a few

minor adaptatio ns, we may put this logical structure into systematic corre-

sp ondence with the mathematical group structure (in the following, A, B,

and C are the group’s operations and E designates its neutral element; a,

b, c , d are objects up on which these oper ations are performed):

i) r e ﬂexivity can be put into correspondence with the existence of the

neutral element E of which the application upon any object le aves the

object unchanged (E exists such that Ea = a)

ii) symmetry can be put into correspondence with the existence of the

inverse operation of which the application on an e lement causes a return

to the original element (for a given A there exists A

−1

such that if

Aa = b then a = A

−1

iii) transitivity c an be put into correspondence with the prop e rty according

to which the product of two of the group’s operations is still an oper-

ation of the group (there exists C such that if BAa = b, then Ca = b,

that is, C = BA)

iv) as for the associative property of the operation products required by

the axiomatic of group structure, it is put into correspondence with the

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160 MATHEMATICS AND THE NATURAL SCIENCES

simple result, associa ted w ith the other properties of the equivalence

relation, according to which [if xRy and yRz and zRu, then xRz and

zRu (applica tion of transitivity) and the second (for the same reaso n)

reduces to xRy and yRu; both are ﬁnally identiﬁed as xRu].

Thus, by this mapping of the respective ax ioms of existence, we highlight

the close formal relationship between the equivalence relation and the group

structure. We may even assert that the mathematical structure of an ab-

stract group constitutes a privileged, non-trivial instantiation of the log ical

equivalence relation. Van Fraassen (1989) goe s even further when asse rting

that “the three concepts of equivalence relation, of partition and of trans-

formation group actually amount to a single concept.” As a matter of fact,

we demonstrate that, as the equivalence clas ses of an equivalence relation

form a partition, there exists a group of which the invariant sets are the

partition’s cells and that each g roup deﬁnes the equivalence relation: “to

be transformed one into the other” for the elements o f the group. Thus, as

far as we are concerned, instead of sp e aking of a single concept applicable

to various ﬁelds, we prefer to consider such a situation as a very strong for-

mal correspondence, an isomorphism, betwe e n distinct c oncepts, each being

adapted to its own disciplinary ﬁeld (sets for mathematics a nd equivalence

relations for logic). Because the important thing is to be able to c onstruct

such a correspondence between the structure of “sensible” symmetries, the

“intelligible” group structure and the “logical” equivalence relation, thus

somewhat comforting the point of view according to which identiﬁcation

and operational use of symmetries stem from very profound cog nitive acts.

Moreover, we have considered that a n essential pa rt of the “identity” of an

object of study co uld be condensed to its symmetry properties (the other

part being mainly an issue of s ymmetry bre aking), it is thus constituted

a sort of epistemological complex which goes from the object’s characteri-

zation linked to the cognitive capac ities of per c e ptio n to that which is the

most formal, re presented by the logic of the equivalence rela tion, and pa ss-

ing by the determination of the relevant symmetries and the mathema tical

group theory.

4.2.1.2 Symmetry breaking, semi-group structure, and order

relations

But if we take the path of the comparison between the mathematical group

structure – an abstract theory of symmetry properties – and the logical

structure o f the equivalence relation, it is then legitimate to question our-

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Invariances, Symmetries, and Symmetry Breakings 161

selves regarding the possibility of ﬁnding similar elements of comparison

for the phenomena associated with symmetry breaking or at least associ-

ated with some of them. In order to attempt to respond to this, we may

notice that while considering earlier the theory of critical phenomena and

the associated symmetry breaking, we had evoked the importance (for their

treatment and the determination of new invariants in terms of universality

classes) of the process of renormalization (Chapter 2, section 2.2.2). Now

the latter appeared to be indissociable from the mathematical structure of

the semi-group, which we k now to be diﬀerentiated fro m the group struc-

ture only by the absence of inverses of the operations which constitute it. It

is then tempting to relate this mathematical structur e of the semi-group to

the logical structure of the order relation which, for its part, diﬀerentiates

itself from the equivalence relation only by the absence of the symmetry

axiom (the (ii) of the equivalence relation), that is, that which we have

put into cor respondence with the ex istence of the inverse operation for the

group axioms. This is how the ﬁliation would be constituted: symmetry

breaking, renormalization semi-gro up, order relation. As the counterpart

of the pr e c e ding sequence (symmetry, group, equivalence relation), this

frame constitutes a rather coherent portrait of the sought correspondences.

Moreover, it relates to our previous analysis of the asymmetry of relatio nal,

irreversible time (the “time arrow”).

4.2.2 Symmetries, symmetry breaking, and determination

of physical reality

We have just been considering a somewhat global approach, founded ﬁrstly

upon the generalized usage of the geodes ic principle within the framework

theories of physics, to the conceptual and theoretical landscape which se e ms

to have progressively organized itself aro und the notions of symmetry and

of symmetry breaking from increasingly abstract constructions. It thus

appears that these concepts constitute another breakthrough in theoretical

generality to a point where, beyond their operational use, some epistemic

principles ﬁnd themselves put into perspective. Let’s b e gin addressing this

point by providing a few examples.

Indeed, a ﬁrst res ulting aspe c t concerns the way in which the concept

of causality is shaken, especially when one attempts to relate it to the

concept of for c e . Thus, general relativity highlights the duality existing

between the characterization of the geometry of the universe and that of

energy-momentum within this universe. By this duality and by the ef-

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162 MATHEMATICS AND THE NATURAL SCIENCES

fects of the invariance principle under the diﬀeomorphisms of space-time,

the “forces” are relativized to the nature of this geometry: they will even

appear or disappear according to the geometrical nature o f the universe

chosen to describe the physical behavior. Likewise for gauge theories, con-

cerning the time internal variables: as in the case of relativity, the choice

of gauges and of their changes enables us to deﬁne certain interactions, or,

conversely, to make them disappear. If we consider that one of the modal-

ities of expression and of observation of the caus al processes is to be found

within the characterization of the forces and ﬁelds which “cause” the ob-

served phenomena, we will see that this modality is thrown into question

by the eﬀects of these transformations. Not that the causal structure itself

is jeopardized, but that the description of its e ﬀects is relativized. This

conduces to formulate a representation of it which is more elabor ate and

more mathematized than the representation resulting from the intuition

stemming fro m classical behaviors. We have already synthesized this a s-

pect in Chapter 1 and recall it here because it is central to our approach:

causes become interactions and these interactions themselves constitute the

fabric of the univers e of their manifestations; reshaping this fabric seems

to modify the interactions, intervening upon the interactions appears to

reshape the fa bric. We will also return to this in the next chapter.

Moreover, concerning the concept o f observability, we have seen that

the existence of invariances under symmetry operations is equivalent to the

expression of formal restrictions to certain possibilities of observation, due

to the absence o f abso lute origins a nd reference systems for space and time.

Thus, gauge theories and relativity groups set the limits to the knowledge

we can obtain about a system and to the quantities we c an meas ure.

What is amazing is that it is precisely these principles of relativity,

these limits imposed upon obse rvation, which contribute to determine,

in a very strong sense, the physical objectivities which conform to them.

These are exactly constituted by invariance properties with respect to non-

absolute reference systems. An analysis of the previously mentioned exam-

ples demo nstrates that an object of fundamental observabilities is always

relative to the framework of reference to which the phenomena are put into

relationship: the space-times of various nature s (including hig hly abstra c t

spaces) within which they ar e deﬁned. Thus, the determination of physical

objectivities seems correlative to the (regulated) indetermination of the ref-

erence systems re lative to which they manifest themselves. As Weyl puts

it: the constructions of objectivity b e gin by choosing a refere nce system

and a measure in it; then, by the analysis of the r e sulting invariances with