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Silviu
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Hello! Can someone explain to me how does a scalar field changes under a Lorentz transformation? I found different notations in different places and I am a bit confused. Thank you!
Thank you for your reply. This makes sense. However I found in Peskin's book on QFT a definition that is different from yours by a prime ( ' ) - I attached a screenshot of it. That is what got me confused. Do you know what does he mean by his notation?vanhees71 said:If ##x'=\Lambda x##, where ##\Lambda## is a Lorentz-transformation matrix, then a scalar field obeys by definition the transformation law
$$\phi'(x')=\phi(x)=\phi(\Lambda^{-1} x').$$
Silviu said:Thank you for your reply. This makes sense. However I found in Peskin's book on QFT a definition that is different from yours by a prime ( ' ) - I attached a screenshot of it. That is what got me confused. Do you know what does he mean by his notation?
Sorry I am a bit confused. What is primed and what is unprimed?George Jones said:The expression that vanhees71 wrote is equivalent to the expression in Peskin and Schroeder, and both expressions are equivalent to
$$ \phi'(Fred)=\phi(\Lambda^{-1} Fred).$$
Why?
What do you mean by rename? If we have a frame S stationary with respect to the field and S' moving with respect to the field (so ##\phi'## and x' are measured in S') then we have by definition ##\phi'(x')=\phi(x)## and by the Lorentz transformation we also have ##\phi(x)=\phi(\Lambda^{-1} x')##. This make sense. But in Peskin he has a mix of both ##\phi'(x)## and this is what confuses me. How does he get to a mix of primed and unprimed indices without a ##\Lambda## factor somwhere? And I am not sure how can you rename x' to x, once you decided which moves and which is fixed. (so to be clear, I understand that choosing prime and unprime as moving or not moving is arbitrary, but Pesking seems to mix them, which confuses me). Thank you for help!vanhees71 said:Just read the equation as a whole. My final equation was
$$\phi'(x')=\phi(\Lambda^{-1} x').$$
Now rename ##x'## back to ##x##, and you get Peskin Schroeder's formula. You can name it "Fred" as suggested in #4 (although that's a bit unusual ;-)).
But the argument of ##\phi'## is always seen from S', right?vanhees71 said:The formula of itself is unique. You can name the argument as you like. You can as well write the law as
$$\phi'(y)=\phi(\Lambda^{-1} y).$$
Of course, the prime at the field symbol on the left-hand side is crucial!
The Lorentz Transformation on Scalar Fields is a mathematical formula that describes how quantities such as time, distance, and energy change when viewed from different reference frames in special relativity.
Understanding Lorentz Transformation on Scalar Fields is important because it helps us make accurate predictions and measurements in the realm of special relativity. It also serves as the basis for many other important concepts, such as time dilation and length contraction.
The Lorentz Transformation affects scalar fields by changing their values in different reference frames. This is due to the fact that scalar fields are quantities that do not have direction, and thus do not transform in the same way as vector quantities under the Lorentz Transformation.
One example of the Lorentz Transformation affecting a scalar field is when we measure the length of an object in two different reference frames. In the reference frame where the object is moving, the length will appear shorter due to length contraction, while in the stationary frame, the length will remain unchanged. This is because the scalar field of length is being transformed by the Lorentz Transformation.
Unlike vector fields, which transform by multiplication with a matrix, scalar fields transform by a simple scaling factor known as the Lorentz factor. This means that the values of scalar fields change in a more straightforward manner when viewed from different reference frames compared to vector fields.