Why is symmetry so important in physics

Continuous symmetries and the Noether theorem

Symmetries form the basic framework on which many physical theories are based. Our present conception of space and time is based on a close interplay of symmetries and fundamental laws of nature. At the center of this knowledge is the so-called Noether theorem.

We encounter symmetries everywhere in everyday life. For example, consider the wonderful rotational symmetry of the rosettes over the entrances to Gothic cathedrals - conceived by the builders as a symbol of the harmony and order of the world created by God. In addition to rotational symmetry, there are also other symmetries of space such as mirror symmetry. We humans look essentially symmetrical when we are reflected in our minds on an axis running from head to foot. The repetitive rhythm of the seasons or a piece of music, in turn, is an expression of a symmetry in time. We are also all familiar with mathematical symmetries: So 1 + 2 = 2 + 1.

Symmetry of the human body

In physics, symmetries often express very simple relationships. But within physical theories they develop a tremendous productivity. In general, a physical system is symmetrical when features of the system remain the same under certain changes. Such changes, such as rotations or reflections, are also called transformations.

Invariance of physical quantities

The German-Jewish mathematician Emmy Noether formulated an example of how at first naive-seeming symmetry requirements generate fundamental laws of physics. It started from the simple requirement that the laws of nature may be the same at any time, in any place and in any direction in the universe.

Of course, the universe doesn't always look the same everywhere, but we assume that the same laws apply everywhere. Physicists also speak of symmetries in this context, the so-called continuous symmetries: It doesn't matter whether you move from any place or point in time in the universe to another - the same laws apply everywhere. For example, if the laws of nature depended on time, you might need less energy today to heat water by ten degrees than the next day, and this difference could generate energy out of nothing. However, such a violation of energy conservation has never been observed.

From the general requirement of the invariance (immutability) of the natural laws of time, place and direction, Emmy Noether derived the theorems of the conservation of energy, momentum and angular momentum, which are fundamental to our understanding of the world. These symmetries of space and time as well as the corresponding conservation laws belong to the foundation of physics today. And so Einstein wrote in his obituary in the New York Times in 1935: "Miss Noether has been the greatest mathematical talent ... since the higher education of women began."

Symmetries go hand in hand with conserved quantities

Emmy Noether recognized a deep connection between the geometric properties of space and time on the one hand and physical quantities on the other. The Noether theorem named after her throws a clear light on the close connection between symmetries and so-called conserved quantities. While quantities such as the position or the speed of an object change in a physical process, the value of a conserved quantity remains constant.

A bouncing ball as an example of energy conservation

A well-known example is the conservation of energy: if you throw a ball in the air, its speed decreases until it comes to a standstill for a brief moment at the highest point before it falls back to the ground. The kinetic energy is converted into potential energy and vice versa, but the total energy remains constant at all times as long as one neglects the friction with the air.

The concept of the conserved quantity is very valuable for physicists, because with constant quantities systems can be described reliably and many calculations can be simplified. In addition to this practical aspect, Noether's theorem also has an aesthetic component: Symmetries are not just “pretty accessories” of nature, but rather a reason why the laws of nature in our world are the way they are.

C, P and T symmetries

But there are other symmetries of space, for example the symmetry against reflections, also called parity or P-operations for short. When physicists discovered in 1956 that the mirror symmetry of space was one hundred percent violated by the weak interaction - one of the four fundamental forces of nature - they were amazed. The deeper reason for this injury is still not understood today.

Parity violation in the weak interaction

Another and somewhat different type of symmetry is the symmetry of our world in relation to the exchange of matter and antimatter, i.e. the exchange of particles and antiparticles. This symmetry is also called charge conjugation or simply C symmetry. Despite parity violation, which was demonstrated, for example, in the so-called Wu experiment, a combination of C and P symmetry, the CP symmetry, initially seemed to be a conserved quantity. But in 1964 came the next shock: The combined CP symmetry is also violated, but only in the per mille range. Building on this, the Soviet physicist Andrei Sacharov explained in 1965 how the observed dominance of matter over antimatter and thus our entire material world can arise through the CP violation from a universe that was originally symmetrical with regard to matter and antimatter.

In 1955, Wolfgang Pauli added the invariance to time reflections, the so-called T-symmetry, to the C and P symmetry. If a system, for example a particle reaction, is invariant under time transformations, it clearly makes no difference to the system whether time runs forwards or backwards.

Pauli introduced the combined CPT transformation and proved the invariance of general field theories under such a joint transformation. Field theories represent a crucial mathematical tool in theoretical physics and describe very fundamentally both classical and quantum physical effects. With the proof that the C, P and T symmetries are not preserved individually, but in combination, the physics was saved! Indeed, to date, no violation of CPT combined invariance has been observed.

In an in-depth article, you will learn which symmetries are hidden in the Standard Model of particle physics.