Midwest Junto for the History of Science
Fifty-fifth annual meeting – March 23-25, 2012
University of Missouri of Science and Technology, Rolla, MO
Symmetry Breaks Out — A Fundamental Concept Jumps Over Disciplinary
Barriers
Eliseo Fernández
Linda Hall Library
[email protected]
Abstract
There are many ways to do research in the history of science, and diverse and oftencomplementary approaches to any given topic. A particular genre of much interest to
philosophers and cognition scientists focuses on the historical genesis and consolidation of
conceptual changes. These events are usually explored by tracing the birth and development of
new concepts as they become articulated in interaction with preexistent background ideas and
with other simultaneously emerging notions.
In this contribution I sketch some highlights in the careers of the concepts of energy, symmetry
and symmetry breaking, concentrating on their role as unifying conceptions of an exceptionally
deep and overarching character. I show how their remarkable explanatory power relates to
their capacity for connecting previously isolated fields of inquiry, first in physics and later in
chemistry and biology.
This narrative proceeds through three principal and consecutive stages. The first concerns the
establishment of the principle of conservation of energy as “the highest law in all science” in the
second half of the 19th century.
The second stage covers the first half of the 20th century and depicts the emergence of the
concept of symmetry —understood in terms of invariance with respect to physical and
mathematical transformations— as the most fundamental conception. I briefly sketch the
central place occupied by symmetry considerations in relativity theory and quantum physics,
and explain how energy conservation assumed a logically subordinate role as a consequence of
its derivation from one of the basic symmetries of nature, through Noether’s theorem.
The final stage is mostly confined to the second half of the last century and sketches the rise of
the concept of symmetry breaking. Its development is outlined from a seminal paper by Pierre
Curie to its present pervasive explanatory function in such previously separated disciplines as
quantum chromodynamics, condensed matter physics and cosmology. Finally, recent
applications of symmetry breaking-ideas to chemistry and biology are noted.
2
Energy and conceptual change
It is the nature of the scientific enterprise to be continuously evolving towards
explanatory accounts encompassing ever-expanding realms of phenomena. To this
purpose scientists construct successive theories and models that unremittingly seek
greater generality as well as deeper unification. In pursuit of these goals they usually
seek to infer both the existence of facts and their explanation from the assumption of a
small set of basic principles and ideas. With the growth of scientific knowledge these
principles and ideas become progressively more basic than those previously
countenanced.
Scientific theories are in permanent need of modification to meet the challenges of
unexplained facts and the emergence of new areas of experience that are made
accessible by the invention of new instruments and the concomitant expansion of
theoretical perspectives. The most important and radical form of these developments is
conceptual change. It consists either in the creation of new concepts or in the
replacement of old ones with modified or otherwise generalized versions. Conceptual
change often proceeds in gradual and incremental fashion, but at times it turns
precipitously during the course of revolutionary episodes, such as those that Thomas
Kuhn famously characterized as “paradigm shifts.”
Kuhn published a justly renowned article on the subject of conceptual change, Energy
Conservation as an Example of Simultaneous Discovery (Kuhn 1959), some four years
before expounding his central ideas on paradigm shifts and conceptual
incommensurability in his classic, The Structure of Scientific Revolutions. In that article
he identified some twelve scientists and engineers who, working largely independently
from one another, managed to come close to a full understanding of the present-day
concept of energy and its conservation (conservation being constitutive of the current
concept of energy, these two notions had to be grasped together).
Since the earliest attempts at reaching a rational understanding of natural phenomena,
something akin to our present concept of energy had been confusedly apprehended
through various notions of a vague and metaphorical character. Those vague
conceptions failed to draw appropriate distinctions from other closely related ideas,
which in time became precisely articulated as our present notions of force, momentum,
power, etc. In the fourth decade of the nineteenth century the successful drawing of
these distinctions became both cause and effect of the emergence of the concept of
energy to its supreme status. Arguably the most central and unifying conception in all of
science, energy rose towards its paramount explanatory role as an exactly definable and
instrumentally measureable quantity — a quantity that is conserved (remains constant)
in all causally isolated physical systems.
In the first half of the nineteenth century there were at least four separate lines of
research that eventually converged at midcentury into the mature, operational concept
3
of energy propounded by Lord Kelvin.1 In this brief treatment I merely touch on Kuhn’s
observations and add considerations drawn from later scholarship.
The first line of research we may consider was a program for the reduction of all
physical phenomena to the action of attractive or repulsive forces between point-like
atoms. To extend this approach beyond the scope of the received Newtonian mechanics
of masses and gravitational forces, a variety of hypothetical, invisible and
"imponderable" fluids had to be countenanced: electrical, magnetic, caloric, etc. Among
the main researchers within this program we shall mention George Green (1793-1841)
and Carl Gauss (1777-1855). The roots of this approach can be traced to eighteenth
century philosophical speculations, in particular to Leibniz’s conception of kinetic energy
as vis viva.
Another parallel tradition, which included experimenters like Jean-Charles Borda (17331799), Charles-Augustin de Coulomb (1736-1806) and Michael Faraday (1791-1867),
sought to describe quantitatively the interactions and transformations of those
hypothetical fluids by inventing and developing sophisticated instruments and
measuring techniques. Their work led to a vague but almost universal conviction of the
reciprocal and universal interconversion of something that was often referred to as “the
forces of nature.”
In France a third and most important research line included a new brand of engineerscientists that flourished under the auspices of the École Polytechnique and other novel
institutions created under Napoleon I. Their efforts, best exemplified by those of Sadi
Carnot (1796-1832), were directed at measuring and improving the efficiency of steam
engines, which were rapidly transforming and expanding manufacture and
transportation in Europe and North America. Engineer-scientists such a James Prescott
Joule (1818-1889), Lord Kelvin (1824-1907) and William Rankine (1820-1872), followed a
similar approach in Great Britain. These investigations led them naturally to the central
concept of work, which they expressed under such rubrics as "mechanical effect."
In Germany a fourth and somewhat separated fourth strand reached the conception of
the conservation of energy via physiological considerations, in the work of Julius von
Mayer (1814-1878) and Hermann von Helmholtz (1821-1894).
In his 1828 work, An Essay on the Application of Mathematical Analysis to the Theories
of Electricity and Magnetism, George Green formally introduced the potential function.
Gauss did the same independently in 1840. We can now recognize the tacit use of this
notion throughout the 18th century, from Euler to Lagrange. The potential describes the
spatial distribution of masses or charges in such a way that the function's rate of change
1
In a parallel development Hermann von Helmholtz articulated the concept of energy with a
similar level of generality in 1847.
4
in the direction of a given point (the gradient in present terminology) yields the value of
the force that would act at that point upon a unit mass or charge.
Green’s work was at first ignored, but Kelvin re-discovered it through a rare copy of the
Essay that he acquired almost accidentally. Years later it led him to being the first to
apprehend clearly the idea of potential energy, defined as the product of the potential
function and the masses or charges involved. In 1851 Rankine formally introduced the
concept of potential energy in terms of the work that is kept stored in the configuration
of a system. Once the relations between work, kinetic energy and potential energy
became defined in both mathematical and instrumental terms, Kelvin was able to fully
articulate the precise concept of energy that we are still using at present, where the fact
of its conservation is often referred to as the first law of thermodynamics (“energy
cannot be created or destroyed”). Through his development of the absolute scale of
temperatures he was also able to lay a solid operational foundation for the second law
of thermodynamics and the role of entropy (a term coined by Rudolf Clausius in 1865) as
a measure of the irreversible dissipation of energy in nature.
Towards the end of the nineteenth century energy (and its conservation) became a sort
of conceptual glue that brought together previously unrelated phenomena and theories,
including mechanics, electromagnetism and heat, under a single explanatory umbrella.
Energy considerations opened up entirely new vistas in biology (e.g., the role of
photosynthesis, respiration and metabolism) and in chemistry (e.g., endothermic and
exothermic reactions).2
Symmetry, invariance, and conceptual change
Just as in the case of energy, vague and pregnant intimations of the idea of symmetry
(and related ideas of harmony, proportion and order) have emerged since ancient times
(see e.g., Darvas 2007, Hon and Goldstein 2008). Symmetry in its current meaning is the
characteristic of a process, system or law that remains invariant (unchanged) after the
performance of an operation or transformation. More precisely stated, symmetry is
invariance under a particular group of transformations, where the term group refers to
a very simple abstract algebraic structure that dictates how the transformations
combine.3 With the benefit of hindsight we now see the current idea already at work in
the discoveries of Galileo (Galilean invariance) and Huygens.
2
The high status of the concept of energy, “the highest law in all science,” was the topic of
expositions in the British popular press in relation to the role of solar energy in the cycles of
nature and the notion of work in human endeavors. See e.g., Underwood 2006.
3
Formally, a group G is a set of elements e with an abstract structure determined by the
properties of a composition operation (“.”) that maps pairs of elements onto new elements. This
composition is associative: e1. (e2 . e3) = (e1 . e2 ) . e3; there is in G a special (neutral) element en,
such that for every e of G we have en . e = e ; and for every e of G there is an inverse element ei
such that e. ei = en .
5
Similarly to the case of energy, symmetry rose to the status of a supreme unifying and
explanatory notion in twentieth-century physics. This development was preceded by the
rise of symmetry to a comparable lofty status in nineteenth century mathematics, in
algebra (e. g., Galois theory) and especially in geometry. Felix Klein’s famous Erlangen
Program sought the unification of the various geometries that arose in the nineteenth
century by placing them in a scheme of increasing levels of generalization, according to
the properties that remained invariant under a hierarchy of different groups of
transformations.
Symmetry first entered the physical sciences in the field of crystallography, in the
classification of symmetric arrangements of atoms. There were other early uses of
symmetry, but the concept’s unparalleled role in contemporary physics came about
through an unprecedented epistemological shift, first fully manifested in Einstein’s
creation of the special theory of relativity in 1905. This move consisted in turning one’s
attention away from the usual business of finding invariances in the phenomena
(discovery of laws) to a consideration of the invariances (symmetries) displayed by the
laws themselves. 4
An essential ingredient of the special theory was the replacement of Galilean invariance,
which tacitly depends on absolute simultaneity, with global Lorentzian invariance, which
belongs to a more general group of transformations under which simultaneity becomes
relative to the state of motion of bodies or systems.5 The most basic postulate of this
theory is the invariance of the laws of physics with respect to all inertial (nonaccelerated) frames of reference. On the other hand, the general theory of relativity,
which Einstein published in 1915, is based on a more general kind of symmetry
(diffeomorphism invariance or general covariance) and has as a basic postulate that the
laws of nature are invariant with respect to all frames of reference (not just inertial
ones). The symmetries involved in classical dynamics and special relativity are global
(invariances under transformations at all spacetime points). Local symmetries
(invariances under transformations that change at different spacetime coordinates)
were to enjoy an even more decisive status in all branches of twentieth century physics.
4
Throughout his career Eugene Wigner repeatedly stated his conviction that the symmetry
principles are at a higher level of generality and that they stand to the laws as the laws
themselves stand to the phenomena (see e.g. Wigner 1985). Marc Lange has greatly clarified the
role of the symmetry principles as “meta-laws” constraining the laws of conservation (see Lange
2007).
5
In Einstein’s terse explanation, the special theory of relativity can “…be summarized in one
sentence: all natural laws must be so conditioned that they are covariant with respect to Lorentz
transformations.” (Einstein 1954, 329).
6
In 1918 a momentous event in the relations of the concepts of energy and symmetry
was marked by the publication of Emmy Noether’s celebrated theorem (Noether 1918).
In non-technical terms it can be stated as a demonstration that (except for some minor
technical reservations) every continuous global symmetry of the laws of nature entails
the existence of a characteristic conserved quantity. For instance, the invariance of the
laws of dynamics under space translations entails the conservation of linear momentum.
Similarly, their invariance under a time translation entails the conservation of energy.
So it turns out that the conservation of energy, the conceptual cornerstone of
nineteenth century physics, becomes in the twentieth century a mere corollary of one
of the global symmetries of the laws of nature. This logical subordination in no way
affects the importance and usefulness of the concept of energy, which remains essential
to physical, chemical and biological explanations.
A special kind of local symmetry, gauge invariance, was of paramount importance in the
development of quantum theory and the subsequent creation of the Standard Model of
particle physics. This model is based in the realization that the fundamental forces of
nature (electromagnetism, the strong nuclear force, etc.) arise from constraints imposed
by gauge symmetries on the laws of nature.
Again in 1918, the year that saw the publication of Noether’s theorem, Hermann Weyl
(1885-1955) discovered the idea of gauge invariance and introduced it in an
unsuccessful attempt at unifying gravitation and electromagnetism. In the 1920’s Weyl
and Eugene Wigner (1902-1995) were among the first physicists to realize the
extraordinary power of symmetry considerations for the development of quantum
theory. Symmetry was at the root of unprecedented conceptual changes through the
creation of new concepts and the discovery of unexpected connections among those
already established. There is no room in this brief account to trace any further these
developments, which came to pass through the protracted work of many of the most
brilliant scientists of all time.
Symmetry breaks away
Since the second half of the twentieth century the notion of symmetry breaking has
risen to center stage and has jumped in unprecedented ways over remotely separated
disciplinary barriers: condensed matter physics, quantum chromodynamics, cosmology,
economics, computer programming, and even string theory and biological evolution.
The easiest way to understand symmetry breaking is through the example of its most
familiar occurrence in phase transitions.
Water, for instance, exists ordinarily in one of three phases: solid, liquid or vapor. The
relations between the phases depend on temperature and pressure. If you have an ice
cube floating in a glass of water at constant atmospheric pressure, as the temperature
increases the ice melts into liquid water and liquid water evaporates. The transitions
from one phase to another are marked by abrupt discontinuities in water density. At
7
constant pressure there are critical values of the temperature at which these
discontinuities occur, such as the freezing point or the boiling point. These
discontinuities mark a breaking of symmetry. The liquid phase is more symmetric than
the solid state. In liquid water all directions are equivalent (rotational invariance) but in
ice a crystal structure with preferred directions emerges, and the rotational symmetry is
broken.
In a seminal article published 1894 Pierre Curie (1859-1906) noted and analyzed the
phenomenon of symmetry breaking without foreseeing, of course, the extraordinary
importance it was to acquire in our understanding of nature in the second half of the
last century. Curie’s principle states that the occurrence of a new phenomenon in a
medium indicates that, through the action of some intervening cause, the original
symmetries of the medium have been reduced to those that are now displayed by the
phenomenon. Such reduction of symmetry is what creates the phenomenon (Curie
1894).
Beginning in the 1960’s the notion of symmetry breaking has unified the once remote
disciplines of particle physics and cosmology. Through the Standard Model, particle
physics seeks to explain the emergence of the various fundamental particles and to
simultaneously unify the fundamental forces of nature. It turns out that the mechanism
that explains the generation of the particles by successive symmetry breakings is the
same as the process that explains the formation of the early universe. Because of those
symmetry breakings, the basic forces (interactions) of nature have very different
characteristics at the low energies prevalent in the present cosmic epoch. But at the
high energies (temperatures) envisaged for the early universe they are expected to
merge into a single force complying with a postulated supersymmetry (SUSY),
describable by a unified gauge theory.
Since mid twentieth century most of the research and applications related to symmetry
breaking have taken place within condensed matter physics, driven by discoveries, such
as the nature of scaling laws and the renormalization group, that can merely be
mentioned here.
Some reflections
In his justly famous and influential article, “More is different,” Philip W. Anderson
(Noble Prize in Physics 1977) offers some speculations on the role of symmetry breaking
in complex systems in general and in living systems in particular:
To pile speculation on speculation, I would say that the next stage could be hierarchy or
specialization of function, or both. At some point we have to stop talking about decreasing
symmetry and start calling it increasing complication. Thus, with increasing complication at each
stage, we go on up the hierarchy of the sciences. (Anderson 1972, p. 396)
Recently, these and similar ideas have begotten several lines of research in such fields as
8
systems biology and evolution. They promise to expand even more the unifying and
generalizing power of symmetry-related concepts. Following Anderson, we note that
concepts acquire new powers as they become embedded into higher levels of a
progressive hierarchical structure. This conceptual hierarchy reflects ascending levels of
complexity in nature. Here again, in the evolution of concepts, we encounter emergence
of novelty at new stages of increasing complexity — more is different.
In this brief sketch I have tried to highlight the ascendancy of concepts toward
increasing generality and unification through the most dramatic example I could find,
the route from energy to symmetry breaking. Similar narratives exist for other
important concepts, and more are waiting to be unearthed. An inventory of case
histories of concept evolution can be of great interest not only to cognition scientists
and philosophers of science, but also to those who want to inject historical depth into
the teaching of scientific ideas and methods.
I cannot think of a better conclusion to these reflections than to reproduce verbatim
Anderson’s conclusion to his paper:
In closing, I offer two examples from economics of what I hope to have said.
Marx said that quantitative differences become qualitative ones, but a dialogue
in Paris in the 1920's sums it up even more clearly:
FITZGERALD: The rich are different from us.
HEMINGWAY: Yes, they have more money.
There is a movie about this.
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