# Metric Sign Convention: Effects on Klein Gordon & Dirac Equations

• iangttymn
In summary, there are two standard conventions for the Minkowski metric, diag(1,-1,-1,-1) and diag(-1,1,1,1), which result in the same physics. However, there are several differences in the equations and algebra depending on which convention is used. Some of these include the form of the Klein Gordon equation, the signs of the Clifford algebra, and the interpretation of time translations. Other changes may also exist, but their physical significance is not fully understood at this time.
iangttymn
The two standard conventions for the Minkowski metric are diag(1,-1,-1,-1) and diag(-1,1,1,1). The physics comes out the same either way, but I'm trying to make a list of the things that change depending on the convention you use.
The Klein Gordon equation is one - with the "mostly plus" (MP) metric it is
$$(\partial^2 - m^2)\phi = 0$$
and with the "mostly minus" (MM) metric it is
$$(\partial^2 + m^2)\phi = 0$$
Another is the sign regarding which is the "positive frequency" solution to Klein Gordon/Dirac. Another is the sign on the Clifford algebra. For MP the more natural choice is
$$\{\gamma^{\mu}, \gamma^{\nu}\} = -2\eta^{\mu\nu}$$
and for MM the more natural choice is
$$\{\gamma^{\mu}, \gamma^{\nu}\} = 2\eta^{\mu\nu}$$
(you can actually make either choice for either metric but the Dirac equation only has the nice "square root" of the Klein Gordan equation form with these choices.

Can anyone point out some other things that are affected by the convention?

Physically, the reversal of the sign means that we reverse the axes and the direction of time. If anything changes at all with the conventions, it must be in an anisotrophic space.

I don't mean any changes in the fundamental physics. I just mean arbitrary conventional changes, like the ones I mentioned. The Klein Gordon equation is saying exactly the same thing in both cases - you just have to write it differently. I'm just trying to get a feel for the different things that are written differently based on the decision.

Real Clifford algebras Cl(1,3) and Cl(3,1) are not isomorphic (complexified are isomorphic). Whether it may have any physical significance is, to my knowledge, not known. I suspect it may, but not in any of today's theories that I know.

Eynstone said:
Physically, the reversal of the sign means that we reverse the axes and the direction of time.
Isn't that wrong? Signature convention is different to swapping the future and past null cones.

I don't understand why anyone chose the mostly negative convention? Mostly positive is maximally consistent with ordinary Riemannian (spatial) geometry, and is perhaps even motivated by incorporating time as the imaginary component..

Mostly negative is being liked in QFT where time translations generator is being interpreted as "energy operator", and "energy" should be "positive". With mostly negative you do not have to flip the sign when you go from covariant $$P_0$$ to contravariant $$P^0$$.

## What is the metric sign convention?

The metric sign convention is a convention used in physics and mathematics to describe the geometry of spacetime. It specifies how distances and intervals are measured and how the components of vectors and tensors are defined.

## How does the metric sign convention affect the Klein Gordon equation?

The Klein Gordon equation is a relativistic wave equation that describes particles with spin 0. The metric sign convention affects this equation by determining the sign of the metric tensor, which is used to define the spacetime interval. This sign affects the solutions of the equation and can lead to different interpretations of the results.

## What is the significance of the metric sign convention in the Dirac equation?

The Dirac equation is a relativistic wave equation that describes particles with spin 1/2, such as electrons. The metric sign convention is crucial in this equation as it determines the sign of the metric tensor and affects the solutions of the equation. This can have implications for the physical interpretation of the results.

## How does the metric sign convention differ from the Minkowski metric?

The Minkowski metric is a specific type of metric used in special relativity, which follows the metric sign convention of (-1,1,1,1). However, the metric sign convention is more general and can vary depending on the context and conventions used. Other common conventions include (+,-,-,-) and (-,-,-,-).

## What are some common applications of the metric sign convention?

The metric sign convention is used in many areas of physics and mathematics, including general relativity, special relativity, and quantum field theory. It is especially important in understanding the geometry of spacetime and the behavior of particles at high speeds and energies.

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