Relationship between levi civita and kronecker delta
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In order to prove the following identity:
∑kϵijkϵlmk=δilδjm−δimδjl" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∑kϵijkϵlmk=δilδjm−δimδjl∑kϵijkϵlmk=δilδjm−δimδjl
Instead of checking this by brute force, Landau writes thr product of Levi-Civita symbols as:
ϵijkϵlmn=det|δilδimδinδjlδjmδjnδklδkmδkn|" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">ϵijkϵlmn=det∣∣∣∣δilδjlδklδimδjmδkmδinδjnδkn∣∣∣∣ϵijkϵlmn=det|δilδimδinδjlδjmδjnδklδkmδkn|
The proof that the equalty holds is quite straightforward if you consider what values the indices can take. But I've been told that there's a much more profound and elegant demonstration based on the representation of the symmetric group.
∑kϵijkϵlmk=δilδjm−δimδjl" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∑kϵijkϵlmk=δilδjm−δimδjl∑kϵijkϵlmk=δilδjm−δimδjl
Instead of checking this by brute force, Landau writes thr product of Levi-Civita symbols as:
ϵijkϵlmn=det|δilδimδinδjlδjmδjnδklδkmδkn|" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">ϵijkϵlmn=det∣∣∣∣δilδjlδklδimδjmδkmδinδjnδkn∣∣∣∣ϵijkϵlmn=det|δilδimδinδjlδjmδjnδklδkmδkn|
The proof that the equalty holds is quite straightforward if you consider what values the indices can take. But I've been told that there's a much more profound and elegant demonstration based on the representation of the symmetric group.
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