In previous posts, I’ve discussed how important nature’s symmetries are to modern physics. So critical, in fact, that Nobel laureate PW Anderson wrote in his widely read 1972 article *More is Different* that “it is only slightly overstating the case to say that physics is the study of symmetry.”

“It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today.” [2]

According to quantum field theory, nature’s playbook starts with symmetry and then connection fields and then forces associated with those fields. Embedded in that matrix are conservation laws. Early in school we heard about these in science class. There’s a deeper story, however, than characterizing collisions on a billiard table.

Famous theoretical physicist Robert L. Mills (co-writer of the Yang-Mills theory) explained it this way in his classes: “For every **conservation law**, there is a **symmetry**. For every symmetry, there is a force field. For every **force field**, there is a **conservation law**.”

Wiki: A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the “transport” of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity which flows in or out of the volume. From Noether’s theorem, each conservation law is associated with a symmetry in the underlying physics [a differentiable symmetry of nature].

Conservation laws are fundamental to our understanding of the physical world, in that they describe which processes can or cannot occur in nature.For example, the conservation law of energy states that the total quantity of energy in an isolated system does not change, though it may change form.

So, let’s summarize some of these relationships in a table [draft version].

Symmetry/invariance | Conservation law | Domain |

Translation in space (x,y,z location)^{1} | Conservation of momentum | Classical physics, QFT |

Time invariance (translation in time) | Conservation of energy | Classical physics, QFT |

Rotation in space (about x,y,z axes)^{2} | Conservation of angular momentum | Classical physics, QFT |

Reflection in space^{3} | Conservation of parity | QFT^{5} |

Gauge invariance | Conservation of electric charge | Classical physics, QFT |

Conservation of lepton number | QFT^{5} | |

Conservation of baryon number^{4} | QFT^{5} | |

Lorentz invariance | CPT symmetry^{6} | |

- Homogeneity of space
- Isotropy of space
- Mirroring, conservation for measured values. “Many physicists reserve the term ‘parity transformation’ exclusively to
**spatial inversion**.” - For nearly all the interactions of the Standard Model.
- Approximately true in particular situations?
- Combining charge, parity and time conjugation.

Wiki:In physics, a **conservation law** states that a particular measurable property of an **isolated** physical system does not change as the system evolves over time. **Exact conservation laws** include conservation of energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. There are also many **approximate conservation laws**, which apply to such quantities as mass, parity, lepton number, baryon number, strangeness, hypercharge, etc. These quantities are conserved in certain classes of physics processes, but not in all.

Also helpful is the book **Symmetries and Conservation Laws in Particle Physics** by Stephen Haywood. A quite technical book. Below is a quote from an excerpt: Chapter 1 “Symmetries and Conservation Laws.”

By definition, the system is said to have a

symmetryif the Hamiltonian is invariant, i.e.H’ = H. Note: this is a symmetry of the Hamiltonian, not of the vector space (Hilbert space) of solutions {⍦}. It isHwhich defines the dynamics of the system, i.e. the interactions. Of course, the symmetry contained withinHwill be reflected in the individual solutions.To summarize: if the

Hamiltonianof a system isinvariantunder aunitary transformationUgenerated by a Hermitian operatorX, then there will be aconserved observableassociated withX. Some of the transformations and their corresponding conserved observables in the case that the Hamiltonian is invariant are listed in Table 1.2.

See also this **American Journal of Physics** (Vol. 72, No. 4, pp. 428–435, April 2004) article “Symmetries and conservation laws: Consequences of Noether’s theorem.”

We derive conservation laws from symmetry operations using the principle of least action.These derivations, which are examples of Noether’s theorem, require only elementary calculus and are suitable for introductory physics. We extend these arguments to the transformation of coordinates due to uniform motion to show that a symmetry argument applies more elegantly to the Lorentz transformation than to the Galilean transformation.

“It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today” (Steven Weinberg). Many of us have heard statements such as

for each symmetry operation there is a corresponding conservation law. The conservation of momentum is related to the homogeneity of space. Invariance under translation in time means that the law of conservation of energy is valid. Such statements come from Noether’s theorem, one of the most amazing and useful theorems in physics. …Symmetries limit the possible forms of new physical laws.The deep connection between symmetry and conservation laws requires the existence of a minimum principle in nature:the principle of least action.

References

[1] http://www.eftaylor.com/pub/symmetry.html

For “symmetry under **uniform linear motion**, known in classical mechanics as **Galileo’s principle of relativity,** … classical action is not invariant under a Galilean transformation.” It can be demonstrated “that the corresponding conservation law to Galilean transformation is related to the uniform **motion of the center of**** ****mass**. … the Galilean transformation and Newton’s laws are only approximate laws of motion. Symmetry under uniform linear motion is a basic assumption of **Einstein’s special relativity”** [which uses the relativistic Lorentz transformation]. …Noether’s theorem can be used … in special relativity to yield the laws of conservation of relativistic energy, momentum, and angular momentum …”

[2] Cited in [1] from R. P. Feynman and S. Weinberg, **Elementary Particles and the Laws of Physics** (Cambridge U.P., Cambridge, 1999), p. 73.

[3] Florida A&M University College of Engineering, Quantum Mechanics for Engineers, Chapter 7.3 “Conservation Laws and Symmetries“

[4] **The Feynman Lectures on Physics**, Volume III Quantum Mechanics, Lecture 17 “Symmetry and Conservation Laws“

Modern physics relies on an underlying premise regarding symmetries — in particular, Noether’s theorem — that space-time is a

continuum. Otherwise, continuous translation and differentiable symmetry are not possible.In this YouTube video of David Tong’s lecture “Quantum Fields: The Real Building Blocks of the Universe”

^{1}published on Feb 15, 2017, in The Royal Institution’s channel, he notes regarding continuum:[1] According to our best theories of physics, the fundamental building blocks of matter are not particles, but continuous fluid-like substances known as ‘quantum fields’. David Tong explains what we know about these fields, and how they fit into our understanding of the Universe. Tong is a professor of theoretical physics at Cambridge University, specialising in

quantum field theory.