Concepts of Symmetries

1. Steps of generalisation of the concept of symmetry

The term symmetry has Greek origins. The word συμμετρια has been formed from συμμ + μετρ(ι)ος and it means in English 'the common measure of things'. See the explanation of the term symmetry in more details in annex [1]. For the Greeks it meant the harmony of parts, proportion, rhythm. The term changed its meaning during the centuries. The first great transformation of the meaning lead from harmony, proportion, rhythm to the geometrical symmetries, like reflection, rotation, translation, glide reflection, similitude, affine projection, topological symmetry, etc. The geometric meanings of symmetry were in close relation to the development and application of symmetry in 19th c. crystallography. The geometrical concept of symmetry had certain limits. At the same time it made possible, that later the symmetries of abstract properties could be generalised according to geometric analogies, based on F. Klein's Erlangen program.

For the sake of generalisation of the concept of symmetry let's first investigate, what was the common in the mentioned geometrical symmetries? In each case

  • one performed a certain (geometrical) operation -> transformation, under which
  • one or more (geometrical) properties of the shape did not change (were conserved).
  • Then this property proved to be invariant under the given transformation.

This geometric interpretation reduced symmetry to the invariance of certain geometric properties of geometric objects under geometric transformations.

Now we can formulate the aspects of generalisation of the geometrical concept of symmetry in the following:

In a generalised meaning, there is symmetry if

  • under any (not certainly geometrical) transformation (operation)
  • at least one (not certainly geometrical) property
  • of the given (not certainly geometrical) object remains invariant (does not change).

Thus, we could extend the meaning of symmetry in physics and beyond. Physics helped to generalise this concept in 3 aspects:

  • for any transformation,
  • for any object, and
  • for any property.

Physics played a distinguished role in this process of generalisation. The introduction of non-geometric, dynamic symmetries brought a new sight to the material structure of our world. The Erlangen program helped to apply geometrical analogies to dynamic symmetries. We learned, how to project, and represent non-geometric properties and their fields in abstract spaces with geometric properties (cf. e. g., the isotopic spin rotation [12]).

2. Classification of symmetries in physics

To understand the role of symmetry played in physics, one should have a back sight to the roots of physics. There were two roots, where our modern physical knowledge originates.

One of them originates in the 17th c., starting with Galileo's relativity principle, which announced the invariance of the laws of motion under the choice of different inertial systems. F. Bacon's generalisation followed this line, which declared the reproductivity of our experiments, and the space- and time-independence of the laws of nature. The investigations of Descartes, Newton and Leibniz continued to develop science in this direction. They were looking for laws of nature and they based their studies on the concept of force.

Another - more or less independent - line of physics originates from the 18th c. variational principles, which were elaborated by Fermat, D'Alembert, Maupertuis, Euler, Jacobi, and finally generalised by Hamilton. Their variational principles were extended later with two more by Gauss, and then Hertz.

If we interpret modern physics in its relation to its origins, one can draw a sharp caesura between the methods of the Newtonian line and modern physics, while the latter line seems a continuous follow up of the Hamiltonian methods. Modern physics justifies the variational principles' origins. We could understand the essence of this choice in the light of the Noether theorems [6].

We must remind, that these two theorems - which play the central role in nowadays physics, and based the role of symmetries in our knowledge on the fundamental structure of matter - were born in the school of D. Hilbert, on the advice of F. Klein, inspired by the emergence of the general relativity theory, and were immediately applied to the laws of conservation by H. Weyl [10,11] and later by E. P. Wigner.

The latter two determined the mathematical foundations of the application of symmetries in 20th c. physics. Both started their career in Germany, then they wrote independently their books on the application of group theory in physics, and later both emigrated in Princeton, where their works became interwoven. Being the younger, Wigner was given the fortune to derive the generalised conclusions: to classify physical symmetries [13].

Wigner classified the symmetries into geometric and dynamic ones. At first glance the basic features of the two kinds of symmetries could be listed in table 1.

geometric symmetries dynamic symmetries
does not change the identity changes identity
laws of nature invariance principles
applies: to (correlation among) events to (the structure of) the laws of nature
global gauge
non-local local
Noether I Noether II

Table 1. Classification of symmetries (1)

This means, that geometric symmetries do not change the identity of the transformed object, that remains the same under transformation (e.g., a boost, a rotation, etc.). Dynamic symmetries in many cases change the identity of the respective object (e.g., transform a proton into a neutron, a red quark into a blue, etc.). Geometric symmetries apply to the correlations among events, while dynamic ones to the laws of nature. In most cases geometric symmetries are global and non-local, while dynamic ones mark gauge, and local invariances.

Geometric symmetries may in many cases described by Noether's 1st theorem, and dynamic ones by Noether's 2 nd theorem [3].

However, this classification is not absolute. There are crossings among the elements of the two columns in the table. The basis, where we can fix our definition of the classification may be that, given by Wigner: what level they do apply. Do they characterise correlations between the physical events or between the laws of nature [13]?

(One should remark, that Hambidge introduced a term 'dynamic symmetry' in mathematics in the twenties, however, the same terminology applied by Wigner differs essentially from that usage.)

In certain cases there are crossings between the properties listed in the two columns of the table. An example for coupling of a gauge invariance and geometric symmetries is the introduction of vector potential in the theory of gravitation. Another crossing, that globality does not certainly coincide with the lack of gauge. E.g., certain ways of derivation of the charge conservation from Noether's 1st theorem [10,11,7].

Therefore we should draw a more differentiated picture on the classification of symmetries, c.f table 2. This classification is based on the differences between the Hamilton principle and Noether's variational problem [2].

This classification is based on the property, to which invariance is subject the given symmetry. The essence of Utiyama's interpretation is in its generalisation, which roots in the different way of derivation of the theorems. This led to 3 sets of relationships (in spite of the two by Noether theorems) and this third group of relationships carries that plus, what extends the applicability of the original theorems [8,9].

Noether I Noether II Utiyama
applies to symmetries associated with finite dimensional Lie groups applies to symmetries associated with infinite dimensional Lie groups generalisation of the Noether theorems
global symmetries local gauge symmetries
continuous symmetry depend on constant parameters transformations, that depend on arbitrary functions and their derivatives 3 sets of relationships +

Table 2. Classification of symmetries (2)

This classification made possible to distinct also transitional cases. These are such so called 'strong' conservation laws, which can be derived independently of any equation of motion, partly from Noether I, partly from Noether II, more precisely: by the application of Noether's 1st theorem to global subgroups of local gauge groups. Thus can we get a global gauge symmetry. Such a symmetry is e. g., Wigner's little group.

Local gauge transformations play the most important role in contemporary physics.

They are transformations that depend on arbitrary functions of space and time. Gauge transformations in field theory are interpreted as transformations of the fields (i.e., the dependent variables) only

Ψi(xμ) → Ψ'i(xμ) (and their derivatives)

and not of the space-time co-ordinates (the independent variables)

xμ → x'μ .

According to Noether II: there are dependencies between the Lagrange expressions and their derivatives. (This follows from the local gauge invariance of the Lagrangian.) Assuming that the gauge field equations are satisfied independently of whether the matter field equations are satisfied: the conserved current expresses the dependence of the matter and gauge fields. In locally gauge invariant theory, satisfaction of the matter field equations is merely a sufficient condition for deriving the existence of a conserved current, and not a necessary condition [2]. Thus we can classify the consequences of the Noether theorems, given in table 3.

According to this interpretation of the theorems Noether grouped the resulting symmetries in the way shown in table 4 [2].

In this light, e. g., the above mentioned crossing between the first classification columns (transitional case of the 'strong' conservation laws), i.e., the derivation of the conservation of the electric charge from the Noether I theorem provides an improper, global gauge symmetry.

Noether I Noether II

Current continuity equation is obtained as a consequence of the matter field equations via N1

Current continuity equation can be derived in conjunction with the gauge field equations via N2

the continuity equation can be derived from the matter field equations,

matter field equations are necessary and sufficient for deriving the continuity equation.

but, the matter field equations are not a necessary condition, they are a sufficient condition for the derivation of the continuity equation.

Consequence:

When the transformations of only the gauge fields depend on the derivatives of the parameters, local gauge symmetry plus satisfaction of the gauge field equations leads to a conserved current.

Table 3. Classification of symmetries (3)

Improper Proper
conservation laws
can be derived without the field equations for the associated field equations being satisfied. when the necessary condition for deriving a conservation law is that the field equations of the associated fields are satisfied.

Table 4. Classification of symmetries (4)

Looking symmetries from an other aspect, both Noether theorems may lead to both Abelian and non-Abelian gauge theories [4,5].

Therefore, as we saw, there are several classification aspects how can we group symmetries. They can be arranged in a matrix according to the following two sets of aspects:

first we can classify them, according to:

  • the conserved quantity,
  • the considered interactions,
  • the kind of their fields (in scalar-vector-tensor and real-imaginary dimensions)
  • the intermediate bosons,
  • particle families,
  • symmetry groups (both simple and combined),
  • the applied gauge (if any),
  • the extent of exactness,
  • ways of description -> dynamics (e. g., QED, QAD, QCD); secondly, in another dimension, we can classify, according to the aspects, which ones from among the characteristics of a symmetry, are considered in each individual case:
  • an invariant property,
  • the considered physical object,
  • the transformation, under which an invariance is found, (one, two, or all the three can be applied).

3. Fulfilling Wigner's program

In his intellectual heritage Wigner left a program for us, to get closer to a deeper understanding of the nature. He appointed symmetries to enlighten our way in this mission.

Are we closer now to more harmony among

  • the phenomena in nature?
    • laws of nature?
      • invariance principles?

and to a GUT or something similar?

Did we get closer to a general law, a general invariance principle or else? We are far from the goal, yet. However, symmetries, and the instructions by Wigner how to use them, proved to be good guides on this way.

Annex - What is symmetry?

Symmetry is a phenomenon, a class of properties and a concept which is present in all scientific disciplines and all kinds of arts

Symmetry bridges different disciplines, sciences and arts and different cultures

The term symmetry is of ancient Greek origin. Its meaning is in close association with the related terms of asymmetry, dissymmetry, antisymmetry. Symmetry and the lack of symmetry characterise the phenomena in our natural and artificial environment, as well as our ideas about the world.

Traditional meaning of symmetry

The meaning of this term went through a fabulous transformation during its use for dozens of centuries. The proper translation of the Greek term symmetria – (from the prefix sym [common] and the noun metros [measure]) – is 'common measure'. The Greeks interpreted this word, as the harmony of the different parts of an object, the good proportions between its constituent parts. Later this meaning was transferred to e.g., the rhythm of poems, of music, the cosmos ('well-ordered system of the universe as contrast of chaos'). Therefore the Latin and the modern European languages used its translations like harmony, proportion until the Renaissance. In wider sense, balance, equilibrium belonged also to this family of synonyms. Some way symmetry was always related to beauty, truth and good. (These relative-meanings determined its application in the arts, the sciences, and the ethics, respectively.) Symmetry was not only related to such positive values, it became even a symbol of seeking for perfectness.

Common meaning of symmetry

In its everyday use symmetry is associated with its most frequent manifestations, like reflection or, in other words, mirror-symmetry, rotation (rotational symmetry), and repetition (translational symmetry). A few further geometrical appearances of symmetry belong also to this class of interpretations, like glide reflection, similitude, affine projection, perspective, topological symmetry . All they are associated with the observation, that one performed a certain geometric operation (a transformation) on an object; and during that transformation one (or more) geometric properti(es) of that geometric object did not change (were conserved). That/those property/ies proved to be invariant under the given transformation. They are called 'symmetry' in everyday life.

Generalised, contemporary meaning of symmetry

In generalised meaning one can speak about symmetry if, (1) under any (not certainly geometric) kind of transformation (operation), (2) at least one (not certainly geometric) property, (3) of the (not certainly geometric) object is left invariant (intact).

Thus we made a generalisation in 3 respects: to
        – any transformation,
        – any object, and its
        – any property.

This generalised meaning of symmetry made possible to apply symmetry to materialised objects in the physical and the organic nature, to products of our mind, etc. Over geometric (morphological) symmetries, we can discuss functional symmetries and asymmetries (e.g., in the human brain), gauge symmetries (of physical phenomena); properties, like colour, tone, shadiness, weight, etc. (of artistic objects).

Asymmetry The lack of symmetry
Dissymmetry The observed object is symmetric in its main features, but this symmetry is slightly distorted (e.g., an arabesque ornament)
Antisymmetry The observed object is symmetric in one of its properties, but one of its other properties changes to its opposite (e.g., a chess-board)

References

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5. Ne'eman, Y. Symmetry: Culture and Science 1 (1990) 229.

6. Noether, E. A., (1918) Invariante Variationsprobleme, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen: Mathematisch-physikalische Klasse, 235.

7. Pauli, W. (1941) Rev. of Mod. Phys. 13, 203.

8. Utiyama, R., (1956) Phys. Rev. 101(5), 1597.

9. Utiyama R., Progr. of Theor. Phys. Suppl. 9 (1959) 19.

10. Weyl, H. (1918) Gravitation und Elektrizität, Sitzungsberichte Preussische Akademie der Wissenschaften, 465-480.

11. Weyl, H. (1929) Electron and gravitation (Engl. transl. in: O'Raifeartaigh (1997): The Dawning of Gauge Theory, Princeton Univ. Press).

12. Yang, C.N., Mills, R.L. (1954) Phys. Rev. 96(1), 192.

13. Wigner, E.P. (1967) Symmetries and Reflections, Scientific Essays of Eugene P. Wigner, Bloomington: Indiana University Press.