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A Why I missed a minus sign?

  1. Nov 24, 2016 #1
    I am reading GEOMETRY, TOPOLOGY AND PHYSICS writen by MIKIO NAKAHARA (second edition). I have a problem on page 400.
    I wonder why the sign is minus in Eq. (10.85).

    And the same problem appears on page 405 in Eq.(10.108). I think it should be minus one half.

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  2. jcsd
  3. Nov 24, 2016 #2


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    Have you considered that with the standard definition (at least in the HEP community) ##\epsilon^{\mu \nu \rho \sigma}=\text{sign}(\mu,\nu,\rho,\sigma)## the covariant components of the Levi-Civita (pseudo-)tensor reads
    $$\epsilon_{\alpha \beta \gamma \delta}=\eta_{\alpha \mu} \eta_{\beta \nu} \eta_{\gamma \rho} \eta_{\delta \sigma} \epsilon^{\mu \nu \rho \sigma}=\det \eta \epsilon^{\alpha \beta \gamma \delta}=-\epsilon^{\alpha \beta \gamma \delta},$$
    because ##\eta=\mathrm{diag}(1,-1,-1,-1)## (or in the east-coast convention ##\mathrm{diag}(-1,1,1,1)##).
  4. Nov 24, 2016 #3
    Thank you! I see.
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