This blog post examines the concept and principles of conserved physical quantities, exploring why they play a fundamental role in physics and are essential for explaining various physical laws and phenomena.
In this world, there exist physical quantities capable of explaining diverse natural phenomena, such as the properties and states of motion of countless objects. Scientists use these physical quantities to formulate various laws and quantitatively describe nature. Among these physical quantities, some are used particularly frequently, and these are precisely the conserved physical quantities.
So, what exactly is ‘conservation’ in physics? Physics is fundamentally the science of defining a system and analyzing the changes in physical quantities within that system. Most physical quantities increase or decrease when subjected to specific transformations within an isolated system. A conserved physical quantity is one whose rate of change with respect to time is zero. Let’s understand this easily through an example of conservation from Feynman’s ‘Lectures on Physics’. Suppose there are initially 15 blocks in a windowless room. A child plays with the blocks inside the room and leaves empty-handed after three hours without tidying them perfectly. To check if the number of blocks changed during those three hours, we weigh the blocks the child tidied. The result shows the weight of 12 blocks. However, since the child did not throw any blocks out the window or take any outside the room, the total number should remain unchanged. With proper observation, the remaining three blocks could be found. Physically, this means the room becomes an isolated system where blocks cannot be taken out, the total number of blocks is a conserved physical quantity, and the child’s action of placing blocks can be seen as a transformation. Even if a physical quantity appears not to be conserved, if its change over time is zero, it is considered conserved.
Physicists have strived to mathematically express the conservation laws satisfied by these conserved quantities rigorously. This effort led to the discovery of Noether’s Theorem by the German mathematician Emmy Noether. Noether’s Theorem states: The state of a system can be expressed by a Lagrangian. Integrating this Lagrangian with respect to time yields the action. The core of Noether’s Theorem is that a symmetry in this action corresponds to a conservation law for a physical quantity. Based on its mathematical rigor, Noether’s Theorem could be applied beyond the easily provable laws of energy conservation, linear momentum conservation, and angular momentum conservation in classical mechanics, extending even to field theory.
So, what physical quantities are conserved? There are two types of laws governing conserved quantities. The first are exact laws proven by the aforementioned symmetries. The second are approximate laws that hold only under restricted conditions, such as low-energy conditions or specific interaction conditions. Physical quantities satisfying exact laws include mass-energy, linear momentum, angular momentum, CPT symmetry (charge, parity, time conjugation), electric charge, color charge, weak isospin, and probability. Physical quantities satisfying approximate laws include rest mass, parity, and lepton number. To briefly explain, the mass-energy conservation law is an invariant under time, linear momentum is invariant under translational motion, angular momentum is invariant under rotational motion, and CPT symmetry is invariant under Lorentz transformations.
Why are these conserved physical quantities particularly widely used and important? It is because these conserved quantities are more fundamental physical quantities. Physics is the discipline that explores the fundamental principles of objects. Therefore, by elucidating and studying the origins of things, we can derive various physical quantities and laws through these conserved quantities. For example, the law of action and reaction can be derived from the symmetry of translational motion.
Conserved quantities appear across diverse physics, spanning from classical mechanics to quantum mechanics. To date, the eight conserved quantities described earlier have been identified. However, the Standard Model, the theory describing fundamental particles, is still evolving. As the Standard Model expands, the possibility remains that we may discover new symmetries. Since new symmetries are highly likely to generate new conservation laws, research into conserved quantities is far from complete. As studies in particle physics and the Standard Model progress and advance, we can anticipate the discovery of new conserved quantities.
