In this blog post, we’ll explore the concept and principles of conserved physical quantities and why they play a fundamental role in physics and are essential for explaining many physical laws and phenomena.
There are many physical quantities in the world that can describe various natural phenomena, such as the properties and states of motion of objects, and scientists use them to formulate laws and describe nature quantitatively. Among these physical quantities, there is one in particular that is used a lot, and that is the conserved physical quantity.
So what is conservation in physics? Physics is essentially the study of setting up a system and analyzing how physical quantities change within that system. Most physical quantities will increase or decrease in an isolated system under certain transformations, and a conserved physical quantity is one that has zero change over time. Consider an example of conservation from Feynman’s Lectures on Physics. Suppose 15 blocks are initially placed in a windowless room. A child plays with the blocks inside the room, and after three hours, walks out of the room empty-handed and not perfectly organized. To see if the number of blocks had changed over the three hours, we weighed the blocks that the child had put away. The result was 12 blocks, but since the child did not throw them out the window or take them out of the room, the total number of blocks would not have changed. With proper observation, you should be able to find the other three. To explain this physically, the room where the blocks cannot be taken out is an isolated system, the total number of blocks is the conserved physical quantity, and the child’s act of placing the blocks is a transformation. Even though a quantity may not appear to be conserved at first glance, it is conserved if the amount of change over time is zero.
Physicists have been trying to rigorously express mathematically the conservation laws that these quantities satisfy, resulting in Noether’s Theorem, discovered by German mathematician Emmy Noether. Noether’s theorem states The state of a system can be represented by a Lagrangian, and integrating this Lagrangian over time yields an action. The idea that each symmetry in this action corresponds to one of the conservation laws of physical quantities is the essence of Neuter’s theorem. The mathematical rigor of the theorem has allowed it to be applied in field theory beyond the easily provable laws of energy conservation, linear momentum conservation, and angular momentum conservation in classical mechanics.
So, what are the conserved physical quantities? There are two kinds of laws about conserved physical quantities. The first is the exact law, which is proved by the symmetries mentioned above, and the second is the analogous law, which only holds when we restrict the situation, such as low velocities or certain interaction conditions. The exact laws include mass-energy, linear momentum, angular momentum, CPT symmetry (charge, parity, time conjugation), charge, color charge, weak isospin, and probability, while the analogous laws include rest mass, parity, and lepton number. Briefly, the conservation of mass and energy are invariant with respect to time, linear momentum is invariant with respect to translational motion, angular momentum is invariant with respect to rotational motion, and CPT symmetry is invariant with respect to Lorentz transformations.
Why are these conserved quantities particularly popular and important? It’s because they are more fundamental. Physics is the study of the fundamental principles of matter, so it is the study of the origin of matter, and from these quantities we can derive various physical quantities and laws. For example, the symmetry of translational motion allows us to derive the action-reaction law.
Conserved quantities appear throughout physics, from classical mechanics to quantum mechanics. These are the eight conserved quantities that have been identified so far. However, the Standard Model, the theory that describes elementary particles, is still evolving, and as it expands, we are likely to discover new symmetries. New symmetries are likely to lead to new conservation laws, so the study of conserved physical quantities is far from over. As the study of particle physics and the Standard Model continues and evolves, it is hoped that new conserved quantities will be discovered.
