Grassman number in functional quantization?

In summary, when working with Grassman numbers in calculations, the order of numbers does not change. Some books may define the complex conjugation of a product of two Grassman numbers to reverse the order, similar to Hermitian conjugation of operators. Despite field operators not being Hermitian, the product of two operators could still be Hermitian, similar to the product of two complex Grassman numbers representing Fermi fields. However, it is unclear if stating that the product of two field operators is Hermitian is correct or not.
  • #1
ndung200790
519
0
Please teach me this:
When calculating something with Grassman numbers without changing order of the numbers,then there are nothing different from ordinary numbers.So I think it would be contrary if we define the complex conjugation of a product of two Grassman numbers to reverse the order of products,just like Hermitian conjugation of operators.But in some books,they do that(they treat the quantization Dirac fields using functional method)
Thank you very much in advanced.
 
Physics news on Phys.org
  • #2
It seem to me that despite field operators is not Hermitian,but the product of the two operators could be Hermitian.So product of two complex Grassman numbers that represent the Fermi fields must be like Hermitian operators.But saying that product of two operators of the field be hermitian is correct or not?
 

FAQ: Grassman number in functional quantization?

What is a Grassman number in functional quantization?

A Grassman number is a type of mathematical object used in functional quantization. It is a number that has both bosonic and fermionic properties, meaning it can commute and anticommute with other Grassman numbers. In functional quantization, Grassman numbers are used to describe fermionic fields, which are particles with half-integer spin.

How are Grassman numbers used in functional quantization?

In functional quantization, Grassman numbers are used as the basic building blocks for fermionic fields. These fields are then used to describe the behavior of particles with half-integer spin, such as electrons and quarks. By using Grassman numbers, it is possible to incorporate the principles of quantum mechanics into the mathematical framework of functional quantization.

What is the relationship between Grassman numbers and supersymmetry?

Supersymmetry is a theoretical framework that describes the relationship between fermions and bosons. Grassman numbers play a crucial role in supersymmetry, as they allow for the creation of superpartners, which are particles with identical properties except for their spin. This is important in theories such as string theory, which incorporate supersymmetry to explain the behavior of fundamental particles.

Are Grassman numbers real or imaginary?

Grassman numbers are neither real nor imaginary in the traditional sense. They are considered to be "super numbers" that have both bosonic and fermionic properties. This means that they do not obey the same mathematical rules as real or imaginary numbers, and instead have their own set of algebraic rules.

Can Grassman numbers be visualized?

Unlike real or imaginary numbers, Grassman numbers cannot be easily visualized. This is because they represent abstract mathematical objects that do not have a physical analogue. However, they are a crucial part of the mathematical framework used in functional quantization and can be manipulated and used in calculations just like any other mathematical object.

Back
Top