This is one of the foundational derivations in relativistic quantum field theory:

showing that the Klein–Gordon scalar field transforms consistently under Lorentz transformations and that the theory is Lorentz invariant.

The key idea is:$$\boxed{ \text{Physics must look identical in all inertial frames.} }$$Physics must look identical in all inertial frames.​

For scalar fields, this turns out to be beautifully simple.

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1. Start with the Klein–Gordon Equation

The scalar field satisfies:$$(\partial_\mu\partial^\mu + m^2)\phi(x)=0$$(∂μ​∂μ+m2)ϕ(x)=0

or equivalently$$(\Box + m^2)\phi(x)=0$$(□+m2)ϕ(x)=0

where$$\Box \equiv \partial_\mu\partial^\mu$$□≡∂μ​∂μ

is the d’Alembertian operator.

Using metric signature $(+,-,-,-)$(+,−,−,−),$$\Box = \frac{\partial^2}{\partial t^2} – \nabla^2$$□=∂t2∂2​−∇2

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2. Lorentz Transformations

A Lorentz transformation changes coordinates:$$x^\mu \rightarrow x’^\mu$$xμ→x′μ

with$$x’^\mu = \Lambda^\mu_{\ \nu}x^\nu$$x′μ=Λ νμ​xν

The matrix $\Lambda$Λ satisfies$$\Lambda^T \eta \Lambda = \eta$$ΛTηΛ=η

which preserves the spacetime interval:$$x_\mu x^\mu = x’_\mu x’^\mu$$xμ​xμ=xμ′​x′μ

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3. What Is a Scalar Field?

A scalar field is defined by:$$\boxed{ \phi'(x’)=\phi(x) }$$ϕ′(x′)=ϕ(x)​

This means:

• the numerical value of the field is unchanged
• only the coordinates labeling spacetime points change

This is the defining property of a Lorentz scalar.

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4. Transforming Derivatives

Now derive how derivatives transform.

Using the chain rule:$$\frac{\partial}{\partial x’^\mu} = \frac{\partial x^\nu}{\partial x’^\mu} \frac{\partial}{\partial x^\nu}$$∂x′μ∂​=∂x′μ∂xν​∂xν∂​

Since$$x’^\mu = \Lambda^\mu_{\ \nu}x^\nu$$x′μ=Λ νμ​xν

the inverse transformation is$$x^\nu = (\Lambda^{-1})^\nu_{\ \mu}x’^\mu$$xν=(Λ−1) μν​x′μ

Therefore:$$\partial’_\mu = (\Lambda^{-1})^\nu_{\ \mu}\partial_\nu$$∂μ′​=(Λ−1) μν​∂ν​

or equivalently$$\partial’^\mu = \Lambda^\mu_{\ \nu}\partial^\nu$$∂′μ=Λ νμ​∂ν

So derivatives transform like four-vectors.

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5. Transforming the d’Alembertian

Now examine$$\Box’ = \partial’_\mu\partial’^\mu$$□′=∂μ′​∂′μ

Substitute the transformed derivatives:$$\Box’ = (\Lambda^{-1})^\nu_{\ \mu}\partial_\nu \Lambda^\mu_{\ \rho}\partial^\rho$$□′=(Λ−1) μν​∂ν​Λ ρμ​∂ρ

Using$$(\Lambda^{-1})^\nu_{\ \mu}\Lambda^\mu_{\ \rho} = \delta^\nu_{\rho}$$(Λ−1) μν​Λ ρμ​=δρν​

we get$$\Box’ = \partial_\nu\partial^\nu = \Box$$□′=∂ν​∂ν=□

Thus:$$\boxed{ \Box \text{ is Lorentz invariant} }$$□ is Lorentz invariant​

This is the central result.

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6. Lorentz Invariance of the KG Equation

Now apply this to the field equation:$$(\Box + m^2)\phi(x)=0$$(□+m2)ϕ(x)=0

Under Lorentz transformation:$$(\Box’ + m^2)\phi'(x’) = (\Box + m^2)\phi(x)$$(□′+m2)ϕ′(x′)=(□+m2)ϕ(x)

Since the RHS is zero,$$(\Box’ + m^2)\phi'(x’)=0$$(□′+m2)ϕ′(x′)=0

Therefore the Klein–Gordon equation has exactly the same form in every inertial frame.

So:$$\boxed{ \text{The KG equation is Lorentz invariant} }$$The KG equation is Lorentz invariant​

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7. Lorentz Invariance of the Action

The action is$$S = \int d^4x\, \mathcal L$$S=∫d4xL

with$$\mathcal L = \frac12\partial_\mu\phi\partial^\mu\phi – \frac12m^2\phi^2$$L=21​∂μ​ϕ∂μϕ−21​m2ϕ2

Now check each term.

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Kinetic Term

$$\partial_\mu\phi\partial^\mu\phi$$∂μ​ϕ∂μϕ

is a Lorentz scalar because it contracts two four-vectors.

Like:$$A_\mu A^\mu$$Aμ​Aμ

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Mass Term

$$\phi^2$$ϕ2

is also scalar since $\phi$ϕ itself is scalar.

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Volume Element

Lorentz transformations preserve spacetime volume:$$d^4x’ = d^4x$$d4x′=d4x

for proper Lorentz transformations.

Therefore:$$S’ = S$$S′=S

Thus the action is Lorentz invariant.

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8. Physical Interpretation

The scalar field has:

• no direction in spacetime
• no spin index
• no vector structure

It behaves like temperature distributed through spacetime.

All observers agree on the field value at the same spacetime event.

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9. Contrast with Spinors and Vectors

Scalar:$$\phi'(x’)=\phi(x)$$ϕ′(x′)=ϕ(x)

Vector:$$A’^\mu(x’) = \Lambda^\mu_{\ \nu}A^\nu(x)$$A′μ(x′)=Λ νμ​Aν(x)

Spinor:$$\psi'(x’) = S(\Lambda)\psi(x)$$ψ′(x′)=S(Λ)ψ(x)

where $S(\Lambda)$S(Λ) is the spinor representation of the Lorentz group.

Spinors transform much more subtly.

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10. The Deep Idea

The entire structure of relativistic QFT comes from demanding:$$\boxed{ \text{the action be Lorentz invariant} }$$the action be Lorentz invariant​

That requirement strongly constrains:

• allowed fields
• allowed interactions
• allowed dynamics

For the Klein–Gordon field, Lorentz invariance emerges because:$$\boxed{ \partial_\mu\partial^\mu }$$∂μ​∂μ​

is a spacetime scalar operator.