Understanding Lorentz Transformation on Scalar Fields

[Silviu]

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!

 

[vanhees71]

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]

If $x'=\Lambda x$, where $\Lambda$ is a Lorentz-transformation matrix, then a scalar field obeys by definition the transformation law
$$\phi'(x')=\phi(x)=\phi(\Lambda^{-1} x').$$

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?

 

[George Jones]

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?

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?

 

[Silviu]

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?

Sorry I am a bit confused. What is primed and what is unprimed?

 

[vanhees71]

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 ;-)).

 

[Silviu]

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 ;-)).

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]

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!

 

[Silviu]

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!

But the argument of $\phi'$ is always seen from S', right?

 

[vanhees71]

Yes, sure.