Hi,In Mandl&Shaw, when we calculate the covaiant commutation

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In summary, the conversation discusses calculating the covariant commutation relations for a scalar field, with the result being zero for space-like intervals. The speaker is confused about why this is the case and suggests using a Lorentz transformation to bring the time component to zero in a time-like interval, but this is not possible due to the Minkowski norm being invariant under Lorentz transformations. The other speaker acknowledges understanding after this explanation.
  • #1
lornstone
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Hi,

In Mandl&Shaw, when we calculate the covaiant commutation relations for a scalar field we obtain :
[tex] [\phi(x),\phi(y)]= i\Delta(x-y)=0[/tex]
and the last equality stands if x-y is a space-like interval. But I don't understand why. We know that it is zero if the time component is zero and we also know that delta is invariant under proper Lorentz transformation. I don't see why we can't do the correct lorentz transformation (which bring time to zero) with a time-like interval so that it is also zero.

Thank you!
 
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  • #2


You can't Lorentz transform a time-like interval into a space-like interval. The Minkowski norm
[tex]
t^2 - x^2 - y^2 - z^2
[/tex]
is invariant under Lorentz transformations, so a time-like interval (positive norm) cannot be mapped into a space-like interval (negative norm).
 
  • #3


I got it, thank you!
 

What is the meaning of "covariant commutation" in Mandl&Shaw?

The term "covariant commutation" refers to the mathematical concept of how two operators, which represent different physical quantities in quantum mechanics, behave when they are applied to a given state. It is a measure of the degree to which these operators can be interchanged without affecting the outcome of a measurement.

What is the importance of calculating covariant commutation?

Calculating covariant commutation is important because it helps to determine the compatibility of different physical quantities and their corresponding operators. This information is crucial in quantum mechanics, as it allows us to predict and understand the behavior of particles and systems at the microscopic level.

How is covariant commutation calculated in Mandl&Shaw?

In Mandl&Shaw, covariant commutation is calculated using the commutator, which is a mathematical operation that measures how two operators do not commute. The commutator is defined as the difference between the product of the two operators and the product of the operators in the opposite order.

What is the significance of the commutator in calculating covariant commutation?

The commutator is significant because it provides a quantitative measure of the non-commutativity of two operators. It also allows us to determine the uncertainty in the measurements of two non-commuting physical quantities, as described by Heisenberg's uncertainty principle.

How does the concept of covariant commutation relate to other principles in quantum mechanics?

Covariant commutation is closely related to other fundamental principles in quantum mechanics, such as the uncertainty principle and the principle of superposition. It also plays a crucial role in the development of quantum field theory and the understanding of symmetries in quantum systems.

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