Quantum Theory, propagator and causality, commutator

[binbagsss]

Homework Statement

Question:

To find/ explain why there exists a continuous lorentz transformation that flips the sign for space-like separation but not time-like.

Homework Equations

Signature $(-,+,+...)$

Definition of lorentz transformation:

$x^u=\lambda^u_v x^v$
$\eta_{uv}\lambda^u_p\lambda^v_{\sigma} = \eta_{p\sigma}$

The Attempt at a Solution

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$\Delta \eta_{uv} y^v x^u < 0$ for time-like seperation
$\Delta \eta_{uv} y^v x^u > 0$ for space-like seperation

And I don't really know where to get started.

Many thanks

 

[mark726]

for your question! I am always eager to discuss and explain scientific concepts.

Firstly, let's define what a Lorentz transformation is. A Lorentz transformation is a mathematical equation that describes how a physical quantity, such as space or time, changes when viewed from different reference frames. It was first introduced by the physicist Hendrik Lorentz in the late 19th century to explain the results of the famous Michelson-Morley experiment, which showed that the speed of light is constant in all reference frames.

Now, let's look at the signature $(-,+,+...)$ that you have mentioned in the equations. This signature is known as the Minkowski metric and it is used to define the spacetime interval between two events. In other words, it tells us whether two events are separated by a time-like interval or a space-like interval. A time-like interval is one in which the events are causally connected, meaning that one event can influence the other. On the other hand, a space-like interval is one in which the events are not causally connected, meaning that one event cannot influence the other.

So, why does a Lorentz transformation flip the sign for space-like separation but not for time-like separation? This is because the Minkowski metric has a different sign for time-like and space-like intervals. As you have correctly stated, for time-like separation, the Minkowski metric is negative, while for space-like separation, it is positive. This is due to the way the metric is defined, which is based on the principle of causality.

Now, let's look at the Lorentz transformation equation $x^u=\lambda^u_v x^v$. This equation tells us how the coordinates of an event, represented by $x^u$, change when viewed from a different reference frame, represented by $\lambda^u_v$. The Lorentz transformation is a linear transformation, meaning that it preserves the underlying structure of spacetime. However, it also takes into account the different signs of the Minkowski metric for time-like and space-like intervals. This is why the sign is flipped for space-like separation but not for time-like separation.

To sum up, the existence of a continuous Lorentz transformation that flips the sign for space-like separation but not for time-like separation is a consequence of the way the Minkowski metric is defined and the principles of causality