The Minkowski transformation group (more commonly the Lorentz group and, when translations are included, the Poincaré group) has several important algebraic properties. One of the most important is that it is generally non-commutative.

1. What is a Minkowski Transformation?

A transformation preserves the Minkowski spacetime interval$$s^2 = c^2 t^2 – x^2 – y^2 – z^2.$$s2=c2t2−x2−y2−z2.

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

then the transformation must satisfy$$\Lambda^T \eta \Lambda = \eta,$$ΛTηΛ=η,

where$$\eta = \begin{pmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{pmatrix}$$η=​1000​0−100​00−10​000−1​​

is the Minkowski metric.

All such matrices form the Lorentz group:$$O(1,3).$$O(1,3).

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2. What Does Commutative Mean?

A group is commutative (Abelian) if$$AB = BA$$AB=BA

for all elements $A,B$A,B.

For spacetime transformations this would mean:

Performing transformation A followed by B gives the same result as performing B followed by A.

This is generally false for Lorentz transformations.

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3. Rotations Are Already Non-Commutative

Consider ordinary 3D rotations.

Rotate first about x-axis and then y-axis:$$R_x R_y$$Rx​Ry​

versus$$R_y R_x.$$Ry​Rx​.

The final orientation is different.

Thus$$R_xR_y \neq R_yR_x.$$Rx​Ry​=Ry​Rx​.

Since spatial rotations are a subgroup of the Lorentz group, the Lorentz group cannot be Abelian.

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4. Lorentz Boosts Are Also Non-Commutative

Consider two boosts:

• $B_x(v)$Bx​(v): boost in x-direction
• $B_y(u)$By​(u): boost in y-direction

Applying them in different orders gives different results:$$B_x(v)B_y(u) \neq B_y(u)B_x(v).$$Bx​(v)By​(u)=By​(u)Bx​(v).

The difference is not merely numerical.

The mismatch generates an additional spatial rotation called the:$$\textbf{Thomas-Wigner Rotation}.$$Thomas-Wigner Rotation.

This is a purely relativistic effect.

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Example

Imagine:

1. Accelerate a spaceship in x-direction.
2. Then accelerate in y-direction.

Compare with:

1. Accelerate in y-direction.
2. Then accelerate in x-direction.

The final velocity is not simply different—the final coordinate axes are rotated relative to one another.

Thus boosts do not commute.

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5. Generator View

The Lorentz group generators are:

Rotations

$$J_i$$Ji​

Boosts

$$K_i$$Ki​

They satisfy:$$[J_i,J_j] = i\epsilon_{ijk}J_k$$[Ji​,Jj​]=iϵijk​Jk​ $$[J_i,K_j] = i\epsilon_{ijk}K_k$$[Ji​,Kj​]=iϵijk​Kk​ $$[K_i,K_j] = -i\epsilon_{ijk}J_k$$[Ki​,Kj​]=−iϵijk​Jk​

where$$[A,B]=AB-BA$$[A,B]=AB−BA

is the commutator.

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Important Observation

The last relation says$$[K_i,K_j]\neq 0.$$[Ki​,Kj​]=0.

Two boosts produce a rotation.

This is the mathematical statement that boosts are non-commutative.

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6. Why Does the Minus Sign Appear?

Compare with ordinary rotation algebra:$$[J_i,J_j] = i\epsilon_{ijk}J_k.$$[Ji​,Jj​]=iϵijk​Jk​.

For boosts:$$[K_i,K_j] = -i\epsilon_{ijk}J_k.$$[Ki​,Kj​]=−iϵijk​Jk​.

The minus sign comes from the Minkowski metric signature$$(+,-,-,-).$$(+,−,−,−).

It reflects the fact that time behaves differently from spatial dimensions.

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7. What About Translations?

When spacetime translations $P_\mu$Pμ​ are added we obtain the Poincaré group.

Translations commute among themselves:$$[P_\mu,P_\nu]=0.$$[Pμ​,Pν​]=0.

However,$$[J_i,P_j] \neq 0$$[Ji​,Pj​]=0

and$$[K_i,P_j] \neq 0.$$[Ki​,Pj​]=0.

Therefore the full Poincaré group is also non-Abelian.

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8. Physical Meaning

The non-commutativity of Lorentz transformations is deeply connected to:

• Relativistic velocity addition
• Thomas precession
• Spin-orbit coupling
• The structure of the Dirac equation
• Quantum field theory representations

In fact, spin-½ particles arise because fields transform under non-trivial representations of this non-commutative Lorentz algebra.

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Summary Table

Transformations	Commutative?
Time translations	Yes
Space translations	Yes
Translation + Translation	Yes
Rotation + Rotation	No
Boost + Boost	No
Rotation + Boost	No
Full Lorentz Group	No
Full Poincaré Group	No

The key commutators are$$[J_i,J_j]=i\epsilon_{ijk}J_k,$$[Ji​,Jj​]=iϵijk​Jk​, $$[J_i,K_j]=i\epsilon_{ijk}K_k,$$[Ji​,Kj​]=iϵijk​Kk​, $$[K_i,K_j]=-i\epsilon_{ijk}J_k.$$[Ki​,Kj​]=−iϵijk​Jk​.

The last equation encapsulates the essential non-commutativity of Minkowski spacetime transformations: two non-parallel Lorentz boosts combine to produce a boost plus a rotation.