Levi-Civita symbol and its effect on anti-symmetric rank two tensors

In summary, the equation provided represents a pattern where only the term with matching indices in the Levi-Civita symbol and the Lorentz algebra matrix remains. The properties of the Levi-Civita symbol and the Lorentz matrices, such as antisymmetry and symmetry, can be used to simplify the equation. Additionally, relabeling of dummy indices can be used to match the terms on the right side of the equation.
  • #1
Jason Bennett
49
3
Homework Statement
below
Relevant Equations
below
I am trying to understand the following:

$$
\epsilon^{mni} \epsilon^{pqj} (S^{mq}\delta^{np} - S^{nq}\delta^{mp} + S^{np}\delta^{mq} - S^{mp}\delta^{nq}) = -\epsilon^{mni} \epsilon^{pqj}S^{nq}\delta^{mp}
$$

Where S^{ij} are Lorentz algebra elements in the Clifford algebra/gamma matrices representation.

The pattern I recognize is that, the only term to remain is the one where the two indices of both the delta and the Lorentz algebra matrix are in the same "slot" of the Levi-Civita symbol. Notably, the m and the p of the delta are both in the first "slot" of the L-C symbol, and the n and q are both in the second "slot" of the L-C symbol.

Can someone help me by pointing out which property of the L-C symbols I ought to be using?

Some additional points that may be on the right track

- anti-symmetric times symmetric = 0
- L-C = anti-sym,
- delta = sym (?), and
- the Lorentz matrices = anti-sym
 
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  • #2
Jason Bennett said:
Homework Statement:: below
Relevant Equations:: below

I am trying to understand the following:

$$
\epsilon^{mni} \epsilon^{pqj} (S^{mq}\delta^{np} - S^{nq}\delta^{mp} + S^{np}\delta^{mq} - S^{mp}\delta^{nq}) = -\epsilon^{mni} \epsilon^{pqj}S^{nq}\delta^{mp}
$$

Where S^{ij} are Lorentz algebra elements in the Clifford algebra/gamma matrices representation.

The pattern I recognize is that, the only term to remain is the one where the two indices of both the delta and the Lorentz algebra matrix are in the same "slot" of the Levi-Civita symbol. Notably, the m and the p of the delta are both in the first "slot" of the L-C symbol, and the n and q are both in the second "slot" of the L-C symbol.

Can someone help me by pointing out which property of the L-C symbols I ought to be using?

Some additional points that may be on the right track

- anti-symmetric times symmetric = 0
- L-C = anti-sym,
- delta = sym (?), and
- the Lorentz matrices = anti-sym
It seems to me that the right hand side of the equation is missing a factor of 4.
All you have to do is to relabel some of the dummy indices so that all the terms are the same as in the expression on the right side. You will only need to use the antisymmetry of the LC symbol and of S.
For example, in the first term, you simply have to switch the indices ##m## and ## n ## and then you will have to use ##\epsilon^{nmi} = - \epsilon^{mni}##.
 
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  • #3
nrqed said:
It seems to me that the right hand side of the equation is missing a factor of 4.
All you have to do is to relabel some of the dummy indices so that all the terms are the same as in the expression on the right side. You will only need to use the antisymmetry of the LC symbol and of S.
For example, in the first term, you simply have to switch the indices ##m## and ## n ## and then you will have to use ##\epsilon^{nmi} = - \epsilon^{mni}##.

UGH! you're absolutely right! Thank you so much! Completely forgot that these are dummy indices (I really need to obey upper/lower even in Euclidean because I forgot about summed over indices all the time if i right things as all upper.) Cheers!
 
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Related to Levi-Civita symbol and its effect on anti-symmetric rank two tensors

1. What is the Levi-Civita symbol and how is it used in mathematics?

The Levi-Civita symbol, also known as the permutation symbol, is a mathematical symbol used to represent the sign of a permutation of a set of numbers. It is commonly used in vector calculus, differential geometry, and other branches of mathematics to simplify and generalize equations involving cross products, determinants, and other operations.

2. What is the relationship between the Levi-Civita symbol and anti-symmetric rank two tensors?

The Levi-Civita symbol is closely related to anti-symmetric rank two tensors, also known as skew-symmetric tensors. In fact, the Levi-Civita symbol can be used to define anti-symmetric tensors, where the components of the tensor are equal to the components of the Levi-Civita symbol multiplied by the corresponding components of another tensor.

3. How does the Levi-Civita symbol affect the properties of anti-symmetric rank two tensors?

The Levi-Civita symbol plays a crucial role in the properties of anti-symmetric rank two tensors. It ensures that the tensor is invariant under coordinate transformations, meaning that it will have the same form in any coordinate system. It also allows for the simplification of equations involving anti-symmetric tensors, making them easier to work with in mathematical calculations.

4. Can the Levi-Civita symbol be extended to higher dimensions?

Yes, the Levi-Civita symbol can be extended to higher dimensions. In three dimensions, it is represented by a 3x3 matrix, but in higher dimensions, it is represented by an n-dimensional matrix. The properties and uses of the Levi-Civita symbol remain the same in higher dimensions as well.

5. How is the Levi-Civita symbol used in physics?

The Levi-Civita symbol is used extensively in physics, particularly in fields such as electromagnetism, quantum mechanics, and general relativity. It is used to simplify equations and express fundamental physical laws, such as Maxwell's equations and the Einstein field equations. It is also used in the calculation of physical quantities, such as angular momentum and magnetic fields.

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