# Rigorous Determination of Bosonic and fermoinic commutation relation

• ClubDogo
In summary, bosonic and fermionic commutation relations are different mathematical rules that govern the behavior of particles in quantum systems. Bosonic particles follow the Bose-Einstein statistics and can occupy the same quantum state, while fermionic particles follow the Fermi-Dirac statistics and cannot occupy the same state. These relations are important in understanding the properties and interactions of particles and are rigorously determined using mathematical techniques. They are fundamental properties of particles and do not change. Furthermore, these commutation relations are related to the Heisenberg uncertainty principle through the concept of operators.

#### ClubDogo

Is there a book that explain in a formal way the deduction of symmetry/antisymmetry of bosonic/fermionic wave equation e/o commutation relation? I've often noticed that some people use examples for the introcution, but is there an axiomatic deduction?

Yes, there are several books that explain the rigorous determination of bosonic and fermionic commutation relations in a formal way. One example is "Quantum Mechanics: A Modern Development" by Leslie E. Ballentine. This book provides a comprehensive and rigorous treatment of quantum mechanics, including the derivation of the commutation relations for bosonic and fermionic operators.

Other books that cover this topic include "Quantum Field Theory" by Franz Mandl and Graham Shaw, and "Quantum Field Theory for the Gifted Amateur" by Tom Lancaster and Stephen Blundell. These books also provide a rigorous and axiomatic approach to understanding the commutation relations for bosonic and fermionic operators.

It is important to note that while examples can be useful for introducing the concept of commutation relations, a formal and rigorous understanding of these relations requires a mathematical approach. This involves starting from the basic principles of quantum mechanics and using mathematical techniques such as group theory and operator algebra to derive the commutation relations for bosonic and fermionic operators.

In summary, there are several books that provide a formal and rigorous deduction of the symmetry/antisymmetry of bosonic/fermionic wave equations and commutation relations. These books are useful resources for scientists and students who are interested in understanding the underlying principles and mathematical foundations of quantum mechanics.

## 1. What is the difference between bosonic and fermionic commutation relations?

Bosonic and fermionic commutation relations are different mathematical rules that describe how particles interact with each other. Bosonic particles, such as photons, follow the Bose-Einstein statistics and have a commutation relation of [a, a†] = aa† - a†a = 1, where a and a† are operators. This means that bosonic particles can occupy the same quantum state. On the other hand, fermionic particles, such as electrons, follow the Fermi-Dirac statistics and have a commutation relation of {c, c†} = cc† + c†c = 0, where c and c† are operators. This means that fermionic particles cannot occupy the same quantum state.

## 2. Why is it important to determine the commutation relations for bosonic and fermionic particles?

It is important to determine the commutation relations for bosonic and fermionic particles because they govern the behavior of these particles in quantum systems. By understanding these relations, we can make predictions about the properties and interactions of these particles, which is crucial for many areas of physics, such as quantum mechanics, condensed matter physics, and particle physics.

## 3. How are bosonic and fermionic commutation relations rigorously determined?

The rigorous determination of bosonic and fermionic commutation relations involves using mathematical techniques, such as operator algebra and group theory, to derive these relations from fundamental principles of quantum mechanics. This process involves carefully considering the symmetry and statistics of the particles, as well as the specific system in which they are interacting.

## 4. Can the commutation relations for bosonic and fermionic particles change?

No, the commutation relations for bosonic and fermionic particles are fundamental properties of these particles and do not change. They are a result of the intrinsic nature of these particles, such as their spin and statistics, and are consistent with experimental observations.

## 5. How do bosonic and fermionic commutation relations relate to the Heisenberg uncertainty principle?

Bosonic and fermionic commutation relations are related to the Heisenberg uncertainty principle through the concept of operators. The Heisenberg uncertainty principle states that certain pairs of physical properties, such as position and momentum, cannot be simultaneously measured with arbitrary precision. Operators, which represent these physical properties, have specific commutation relations that are related to the uncertainty principle. For example, the commutation relation [x, p] = xp - px = iħ, where x is the position operator and p is the momentum operator, is a fundamental result of the uncertainty principle.