Grassmann Numbers & Commutation Relations

• RedX
In summary, the conversation discusses the expression <0|\eta|0> and its definition when \eta is a Grassman number that anticommutes with creation and annihilation operators. It is shown that the expression is well-defined and vanishes when \eta is a constant.
RedX
If you have a Grassman number $$\eta$$ that anticommutes with the creation and annihilation operators, then is the expression:

$$<0|\eta|0>$$

well defined? Because you can write this as:

$$<1|a^{\dagger} \eta a|1>=-<1| \eta a^{\dagger} a|1> =-<1|\eta|1>$$

But if $$\eta$$ is a constant, then shouldn't:

$$<0|\eta|0>=<1|\eta|1>=\eta$$ ?

RedX said:
If you have a Grassman number $$\eta$$ that anticommutes with the creation and annihilation operators, then is the expression:

$$<0|\eta|0>$$

well defined? Because you can write this as:

$$<1|a^{\dagger} \eta a|1>=-<1| \eta a^{\dagger} a|1> =-<1|\eta|1>$$

But if $$\eta$$ is a constant, then shouldn't:

$$<0|\eta|0>=<1|\eta|1>=\eta$$ ?

Grassmann numbers are operators (though they are called numbers).
$$<0|\eta|0>=0$$ is well-defined and vanishes.

1. What are Grassmann numbers and how are they different from regular numbers?

Grassmann numbers are mathematical objects used in the field of Grassmann algebra, which is an extension of linear algebra. Unlike regular numbers, they do not commute with each other, meaning that their order matters when performing mathematical operations.

2. What are some applications of Grassmann numbers?

Grassmann numbers have a wide range of applications, including in quantum mechanics, theoretical physics, and differential geometry. They are also used in computer science for data analysis and in economics for modeling financial markets.

3. What are commutation relations and how do they relate to Grassmann numbers?

Commutation relations describe the behavior of operators in quantum mechanics, and they are defined by the order in which operators are applied. Grassmann numbers do not commute with each other, which means they follow a different set of commutation relations than regular numbers.

4. Can Grassmann numbers be used to represent fermions?

Yes, Grassmann numbers are often used to represent fermions in physics. This is because they have the unique property of anti-commutation, which is essential for modeling fermionic behavior.

5. Are there any limitations or challenges in working with Grassmann numbers?

One limitation of Grassmann numbers is that they do not have a well-defined notion of equality, making it challenging to determine when two Grassmann numbers are equal. Additionally, their non-commutative nature can make calculations and manipulations more complicated compared to regular numbers.

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