Metric Signatures

Changing the metric signature changes the sign of contractions, but it does not change a physical energy, propagation direction, or probability. A consistent translation also tracks the phase convention, Clifford relation, adjoint, and source normalization. This lookup applies the accepted Metric and Units ledger to one explicitly defined mostly-plus alternative.

Required background. Metric and Units defines the original convention package. Helpful background. Gamma-Matrix Conventions supplies the matrix definitions, and Klein–Gordon Propagators explains source and boundary factors.

Metric signatures and index translation

Use $\hbar=c=1$ on this page. Keep the same contravariant inertial coordinate and physical momentum components, $x^\mu=(t,\mathbf x)$ and $p^\mu=(E,\mathbf p)$, while comparing

$$\eta=\operatorname{diag}(1,-1,-1,-1), \qquad g=-\eta.$$

Write $p_{\eta\mu}=\eta_{\mu\nu}p^\nu$ and $p_{g\mu}=g_{\mu\nu}p^\nu$. The subscript on a contraction identifies which metric is used.

Quantity	Mostly minus, $\eta$	Mostly plus, $g=-\eta$
Lowered momentum	$(E,-\mathbf p)$	$(-E,\mathbf p)$
Scalar product	$a^0b^0-\mathbf a\cdot\mathbf b$	$-a^0b^0+\mathbf a\cdot\mathbf b$
Massive mass shell	$p_\eta^2=m^2$	$p_g^2=-m^2$
Timelike interval	$ds_\eta^2>0$	$ds_g^2<0$
Wave operator	$\Box_\eta=\partial_t^2-\nabla^2$	$\Box_g=-\partial_t^2+\nabla^2$
Free scalar equation	$(\Box_\eta+m^2)\phi=0$	$(\Box_g-m^2)\phi=0$

The coordinate derivative $\partial_\mu=\partial/\partial x^\mu$ is unchanged. Raising its index uses the metric, so $\partial_g^\mu =-\partial_\eta^\mu$. Similarly, $p_{g\mu}=-p_{\eta\mu}$, although the physical energy component $E=p^0$ is the same positive number. Both mass-shell rows give $E^2=\mathbf p^2+m^2$.

Fix the same coordinate orientation $\epsilon^{0123}=+1$. In four dimensions both displayed metrics have determinant $-1$, so lowering all four indices gives $\epsilon_{0123}=-1$ in both. The words “mostly plus” do not by themselves fix an author’s orientation.

Positive frequency and the source equation

The same positive-frequency wave is

$$e^{-ip_{\eta\mu}x^\mu} =e^{+ip_{g\mu}x^\mu} =e^{-iEt+i\mathbf p\cdot\mathbf x}.$$

Keeping the written minus sign in front of the newly defined contraction would instead reverse the frequency. Translate the phase before applying a momentum substitution or interpreting a pole.

For the scalar operators in the table, $L_g=-L_\eta$. If $L_\eta G_\eta=\delta$ with a specified retarded, advanced, or Feynman boundary prescription, the corresponding unit-source inverse is $G_g=-G_\eta$:

$$L_g(-G_\eta)=\delta.$$

This sign is a source-normalization statement. It does not say that a fixed physical vacuum correlator changes sign when its notation changes. For example, the same Feynman correlator multiplier can be written

$$\frac{i}{p_\eta^2-m^2+i0} =\frac{-i}{p_g^2+m^2-i0},$$

with the corresponding phase translation above. Its source is $-i\delta$ under $L_\eta$ and $+i\delta$ under $L_g$. Dropping the numerator sign or retaining the old sign of $i0$ changes this identity or its boundary prescription. See Klein–Gordon Propagators for the inverse/correlator distinction.

One consistent Clifford and adjoint package

A metric name alone does not determine the displayed Clifford convention. Here choose

$$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}I, \qquad \Gamma^\mu=i\gamma^\mu.$$

Then $\{\Gamma^\mu,\Gamma^\nu\}=2g^{\mu\nu}I$. With the same spinor components, the following package gives the identical free Dirac equation and physical current:

Object	Original matrices	Translated matrices
Free equation	$(i\gamma^\mu\partial_\mu-m)\psi=0$	$(\Gamma^\mu\partial_\mu-m)\psi=0$
Adjoint	$\bar\psi=\psi^\dagger\gamma^0$	$\bar\psi=-i\psi^\dagger\Gamma^0$
Current	$j^\mu=\bar\psi\gamma^\mu\psi$	$j^\mu=-i\bar\psi\Gamma^\mu\psi$
Time matrix	$(\gamma^0)^\dagger=\gamma^0$	$(\Gamma^0)^\dagger=-\Gamma^0$
Spatial matrices	$(\gamma^i)^\dagger=-\gamma^i$	$(\Gamma^i)^\dagger=\Gamma^i$

For this particular dictionary, $i\Gamma^0\Gamma^1\Gamma^2\Gamma^3=\gamma^5$. Other written definitions acquire factors: for example, $\tfrac i2[\Gamma^\mu,\Gamma^\nu] =-\tfrac i2[\gamma^\mu,\gamma^\nu]$. Translate the definitions of tensor bilinears and generators as well; do not simply insert new matrices into an unchanged list of formulas.

Some sources using $g=(-+++)$ instead define their Clifford relation with $-2g^{\mu\nu}$ and retain the original gamma matrices. Their convention is consistent too, but it is not the particular $\Gamma=i\gamma$ dictionary above. Read the source’s actual equation and adjoint rather than inferring them from its metric signature. The table concerns free equations; background-field work must also translate the potential and covariant derivative package in Minimal Coupling.

A basis change keeps the signature

A constant unitary similarity $\gamma^\mu\mapsto V\gamma^\mu V^\dagger$ preserves the original anticommutator:

$$\{V\gamma^\mu V^\dagger,V\gamma^\nu V^\dagger\} =2\eta^{\mu\nu}I.$$

Thus the Dirac and chiral bases are two representations of the same signature. They are not the mostly-minus and mostly-plus alternatives. Use Gamma-Matrix Conventions for basis choices and the spinor notebook for executable similarity and antilinear congruence checks.

Translation checks

Rest wave. After translation, a positive-energy rest solution must still have time dependence $e^{-imt}$. The sign of a written contraction alone is not its frequency label.

Charge density. With the displayed adjoint/current dictionary, $j^0$ must still equal $\psi^\dagger\psi$. An imaginary density signals that a factor from the matrix conversion was omitted.

Scalar source. If the differential operator is multiplied by minus one while the unit source is fixed, its inverse must also change sign. Test this product before assigning a name to the new kernel.

References

• Bjorken, James D., and Sidney D. Drell. Relativistic Quantum Mechanics. McGraw–Hill, 1964.
• Dreiner, Herbi K., Howard E. Haber, and Stephen P. Martin. “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry.” Physics Reports 494, 1–196 (2010). doi:10.1016/j.physrep.2010.05.002; corrected arXiv version, appendix G.