# Klien Gordon Equation solution under Lorentz transformation

• McLaren Rulez
In summary, the conversation discusses the transformation law for the wavefunction in Relativistic Quantum Mechanics and its relation to the Poincare group. The speaker is unsure why the transformation must have the form of \psi'(x')=\lambda \psi(x) with |\lambda|=1 and asks for clarification. The response explains that the scalar field's behavior under Poincare transformations is a result of a quantum theory invariant and the minus sign comes from considering the full Poincare group.
McLaren Rulez
Hi,

I am using Griener's Relativistic Quantum Mechanics and I have a question. Using the Klien Gordon Equation $(p^{\mu}p_{\mu}-m_{0}c^{2})\psi=0$, he says that the transformation law for the wavefunction i.e $\psi(x)$ transforming to $\psi'(x')$ must have the form $\psi'(x')=\lambda \psi(x)$ with $|\lambda|=1$. I don't understand why this is the case. Can anyone help me see why this must be so?

Thank you

It doesn't follow really. I'm thinking that any massive spin n representation of the Poincare group must obey the mass sheet condition: p^2 - m^2 = 0.

Think about the Dirac field for a second: by squaring the Dirac equation you get the Klein-Gordon equation which is satisfied by the field, but under restricted Poincare transformation, the Dirac field transforms like a (1/2,0) directsum (0,1/2) spinor, not like a (pseudo)scalar.

I am a doing this as a bit of holiday reading so I'm actually completely unfamiliar with what you're saying. So basically, how do we know how the wavefunction transforms when we go from one frame to another? What tells us that it must be the same up to a minus sign?

Thank you

[...]So basically, how do we know how the wavefunction transforms when we go from one frame to another? [...]

Actually, we don't. The scalar field's existence is a consequence of a quantum theory invariant under Poincare trasformations. Its behavior under such transformations is again a result, not an assumption. The minus sign comes from considering the full Poincare group, or a subgroup containing the spatial reflection.

Thank you dextercioby

## What is the Klien Gordon Equation?

The Klien Gordon Equation is a relativistic wave equation that describes the behavior of spinless particles, such as scalar fields, in a Lorentz-invariant manner.

## What is a Lorentz transformation?

A Lorentz transformation is a mathematical transformation that describes how the coordinates of an event change between two inertial frames of reference that are in uniform relative motion.

## How does the Klien Gordon Equation change under a Lorentz transformation?

The solution of the Klien Gordon Equation under a Lorentz transformation will have the same functional form, but the coefficients and arguments of the solution will change due to the transformation of coordinates.

## What is the physical significance of the Klien Gordon Equation solution under a Lorentz transformation?

The solution of the Klien Gordon Equation under a Lorentz transformation describes the behavior of a spinless particle in a moving reference frame, allowing us to understand how the particle's properties, such as energy and momentum, change as it moves at relativistic speeds.

## What are some applications of the Klien Gordon Equation under Lorentz transformation?

The Klien Gordon Equation and its solutions under Lorentz transformation have applications in various fields, including quantum mechanics, particle physics, and cosmology. They are used to describe the behavior of scalar particles, such as Higgs bosons, and to understand the effects of relativity on the behavior of particles in different reference frames.

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