# Symmetry of Quantum vs Classical Physics Systems

• KFC
In summary, the author of the introductory book on symmetry in physics explains that the concept of symmetry in quantum mechanics is significantly different from that of classical mechanics, particularly in regards to angular momentum and time reversal. However, both classical and quantum mechanics share the idea that symmetry is tied to the conservation of a generator, with QM being constructed from analogy with CM.
KFC
I read a introductory book on symmetry of physics system. The author says symmetry of quantum mechanical system is VERY different from that of classical system. I just wonder what's the main difference?

KFC said:
I read a introductory book on symmetry of physics system. The author says symmetry of quantum mechanical system is VERY different from that of classical system. I just wonder what's the main difference?

I'd say the main difference is in something like angular momentum. Rotational invariance leads to conservation of angular momentum, but in QM you only know the angular momentum in one direction. There might also be a difference in time reversal, where spin 1/2 particles pick up a negative sign when performed twice or something, but I'm not sure about this.

Other than that, CM and QM are similar with regards to symmetries, and usually QM is constructed from analogy with Hamilton's CM. In CM, symmetry means the Hamiltonian is unchanged by the flow generated by a canonical generator of the symmetry - but there is a dual relationship that says that this implies that the generator is unchanged by the flow generated by time translation (which happens to be the Hamiltonian). So basically if the flow created by a generator does not change the generator of the flow of time, then the flow of time does not change the generator. Hence the generator is conserved. This is true in CM and QM.

which book?

## 1. What is the difference between quantum and classical physics systems in terms of symmetry?

In classical physics, symmetry refers to the invariance of physical laws under transformations such as translations, rotations, and reflections. In contrast, quantum systems exhibit a broader range of symmetries, including discrete symmetries such as charge conjugation and parity, which do not have classical analogues. Additionally, quantum systems can exhibit symmetries that arise from the superposition and entanglement of states, which are unique to the quantum realm.

## 2. How does the concept of symmetry play a role in quantum mechanics?

Symmetry is a fundamental concept in quantum mechanics, as it allows us to classify and understand the behavior of quantum systems. By studying the symmetries of a system, we can predict its properties and behavior, such as energy levels and allowed transitions between states. Symmetry also plays a crucial role in the development of quantum field theories, which are essential for describing the behavior of particles and fields at a fundamental level.

## 3. Can symmetries be broken in quantum systems?

Yes, symmetries can be broken in quantum systems, just as they can in classical systems. In quantum mechanics, this is known as spontaneous symmetry breaking, and it occurs when the ground state of a system does not exhibit the same symmetries as the underlying laws of physics. This phenomenon is crucial in understanding the behavior of many quantum systems, including superconductors and superfluids.

## 4. How does quantum entanglement affect the symmetries of a system?

Quantum entanglement, where two or more particles become correlated in such a way that their properties are dependent on one another, can lead to the emergence of new symmetries in a system. This is because entangled particles exhibit a type of symmetry known as entanglement symmetry, which is fundamental to the study of quantum information and quantum computing.

## 5. Are there any practical applications of studying the symmetry of quantum vs classical systems?

Yes, understanding the symmetries of quantum systems has several practical applications. For example, symmetries play a crucial role in the design of quantum algorithms and protocols for quantum communication and cryptography, which rely on the entanglement of particles. Additionally, studying the symmetries of quantum systems can also provide insights into the behavior of complex systems, such as materials and biological molecules, which have quantum properties at the microscopic level.

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