How to Derive Lorentz Scalar and 4-Vector Equations in QFT?

[MathematicalPhysicist]

I hope to replicate my previous thread in QFT which was started three years ago from reading Srednicki's textbook and solution manual and also the problem book that I read (by some serbian fellow).

This time I am planning to read several books, so the the title of this thread is general.

Anyway, I'll start with my first question from Bjorken's and Drell's classic fields book.

My question refers to pages 153-155:
I'll quote the passage:

As before, we introduce the spectral amplitude by grouping together in the sum over $n$ all states of given four momentum $q$:
$$(16.97)\rho_{\alpha \beta}(q) = (2\pi)^3\sum_n \delta^4(p_n-q)\langle 0 | \psi_\alpha (0) |n \rangle \langle n|\bar{\psi}_\beta(0)| 0\rangle$$
and set out to construct its general form from invriance arguments.
$\rho(q)$ is a $4\times 4$ matrix and may be expanded in terms of the 16 linearly independent products of the $\gamma$ matrices:
$$(16.98)\rho_{\alpha \beta}(q) =\rho(q)\delta_{\alpha \beta}+\rho_\mu(q)\gamma_{\alpha\beta}^\mu + \rho_{\mu\nu}(q)\sigma_{\alpha\beta}^{\mu\nu}+\tilde{\rho}(q)\gamma_{\alpha\beta}^5+\tilde{\rho}_\mu(\gamma^\mu\gamma^5)_{\alpha\beta}$$
...
$$(16.102)\rho(q)=S^{-1}(a)\rho(qa^{-1})S(a)$$
where the matrix $S$ is defined by:
$$(16.100)S^{-1}\gamma^{\mu}S= a^\mu_\nu \gamma^\nu$$
Equation (16.102), together with the general expansion of $\rho(q)$ given in (16.98), determines the structure of the coefficients $\rho, \rho_\mu$, etc. For instance, if (16.98) is inserted into (16.102), it follows that
$$\rho(q)=\rho(qa^{-1})$$
that is, $\rho$ transforms as a Lorentz scalar. Similarly, $\rho_\mu(q)=a_\mu^\nu \rho_\nu(qa^{-1})$ transforms as a Lorentz 4-vector, and so on.
...
In this way the form (16.98) is limited to
$$(16.103)\rho_{\alpha\beta}(q)=\rho_1(q^2)\not{q}_{\alpha\beta}+\rho_2(q^2)\delta_{\alpha\beta}+\tilde{\rho}_1(q^2)(\not{q}\gamma^5)_{\alpha\beta}+\tilde{\rho}_2(q^2)\gamma_{\alpha\beta}^5$$
To reduce the form of $\rho_{\alpha\beta}(q)$ further, we must require invariance of the theory under the parity transformation $\mathfrak{P}$, which has the property:
$$(16.104)\mathfrak{P}\psi_\alpha(0)\mathfrak{P}^{-1} = \gamma_{\alpha \lambda}^0 \pi_\lambda(0)$$
Inserting (16.104) into (16.97) and carrying out the steps analogous to those used for proper Lorentz transformations, we arrive at the analogue of (16.102):
$$(16.105)\rho_{\alpha \beta}(q,q_0) = \gamma^0_{\alpha \beta} \rho_{\lambda \delta}(-q,q_0)\gamma_{\delta \beta}^0$$
Inserting (16.103) into (16.105) one finds
$$(16.106) \tilde{\rho}_1 = \tilde{\rho}_2=0$$

My two questions:

1. How to rigorously derive the two equations for Lorentz scalar and 4 vector in the quote above ?
2. How to rigorously derive equations (16.105) and (16.106)?

Thanks in advance, I plan on reading also Boyarkin's book and Hatchinson's and other books in QFT and QCD.Cheers!

 

[MathematicalPhysicist]

I am reading the book called: "Gauge theory of elementary particle physics" by Ta-Pei Cheng and Ling-Fong Li.

On pages 19-20 they write: $(1.78) W[J] = \bigg[\exp(\int d^4x (\mathcal{L}_1 (\frac{\delta}{\delta J}))\bigg] W_0[J]$, where: $W_0[J]=\int [d\phi] \exp\bigg[ \int d^4x (\mathcal{L}_0+J\phi)\bigg]$.

Now, on page 20 they write:" The perturbative expansion in powers of $\mathcal{L}_1$ of the exponential in (1.78) gives:

$$(1.85) W[J] = W_0[J]\bigg\{ 1+\lambda\omega_1[J]+\lambda^2 \omega_2[J]+\ldots \bigg \},$$

where $$(1.86) \omega_1[J] = -\frac{1}{4!}W_0^{-1}[J]\bigg\{ \int d^4x \bigg[\frac{\delta}{\delta J(x)} \bigg]^4 \bigg\} W_0[J]$$

$$\omega_2[J]=-\frac{1}{2(4!)^2} W_0^{-1}[J]\bigg\{ \int d^4x \bigg[ \frac{\delta}{\delta J(x)}\bigg]^4\bigg\}^2 W_0[J] = $$

$$ = -\frac{1}{2(4!)} W_0^{-1}[J]\bigg\{ \int d^4x \bigg[\frac{\delta}{\delta J(x)}\bigg]^4\bigg\} \omega_1[J]$$
Now, for my question, after I plug $\omega_1[J]$ into the above last equation I get:

$$\frac{1}{2(4!)^2} W_0^{-1}[J]\{ \int d^4 x \bigg[ \frac{\delta}{\delta J(x)} \bigg]^4 \} W_0^{-1}[J] \{ \int d^4 x \bigg[ \frac{\delta}{\delta J(x)} \bigg]^4 \} W_0[J]$$

The last expression is not the same as the above expression, i.e. of $-\frac{1}{2(4!)^2} W_0^{-1}[J]\bigg\{ \int d^4x \bigg[ \frac{\delta}{\delta J(x)}\bigg]^4\bigg\}^2 W_0[J]$.

Perhaps instead of $\omega_1[J]$ it should be $-W_0[J] \omega_1[J]$ in equation (1.86)?
I am puzzled, what do you think?

 

[MathematicalPhysicist]

Anyone care to comment on post #3?

 

[kith]

I don't know the answer to your question, so just a general comment. Your current method of asking has two disadvantages:
1) People who are willing to help can't see what your question is about by reading only the title of the thread. This makes it less likely that you get answers.
2) People who are searching the forums for answers have a harder time to find them if a thread has multiple topics.

So it would be better if you started a new thread for every question (even though I understand the desire to have it all in one place for yourself).

 

[vanhees71]

Anyone care to comment on post #3?

You are right. As it stands it's wrong and must be corrected as you say. There are also some factors $\mathrm{i}$ missing. Perhaps, another book would be better!

My favorite for introductory QFT is

M. D. Schwartz, Quantum field theory and the Standard Model, Cambridge University Press, Cambridge, New York, 2014.

 

[MathematicalPhysicist]

You are right. As it stands it's wrong and must be corrected as you say. There are also some factors $\mathrm{i}$ missing. Perhaps, another book would be better!

My favorite for introductory QFT is

M. D. Schwartz, Quantum field theory and the Standard Model, Cambridge University Press, Cambridge, New York, 2014.

I find that each book, it or its solution manual has some "problems", you can look at my thread that you had participated in it:
https://www.physicsforums.com/threads/another-question-from-srednickis-qft-book.761915/

For the book by Cheng et al there is an errata but they don't specify any errors on pages 19-20.

 

[MathematicalPhysicist]

I don't know the answer to your question, so just a general comment. Your current method of asking has two disadvantages:
1) People who are willing to help can't see what your question is about by reading only the title of the thread. This makes it less likely that you get answers.
2) People who are searching the forums for answers have a harder time to find them if a thread has multiple topics.

So it would be better if you started a new thread for every question (even though I understand the desire to have it all in one place for yourself).

It's not just for myself, in the beginning of physicsforums there were giant threads where you discussed issues such as differential forms, pdes etc and they had quite a lot of pages with interesting topics.
such as this:
https://www.physicsforums.com/threads/intro-to-differential-forms.2953/

I just have a feeling that something is wrong with this framework of QFT if there are so many stuff that is wrong in the derivations in different books.

You would think that after so many books have been written on this topic that eventually there would a perfect book with spotless answers to problems with all the derivations.

I have Ryder's,Brown's,Ramond's, Zuber's& Itzykson's, Peskin's and Schroeder's, Srednicki, Atchinson's, Ticciati's, Folland's and Weinberg's; these books seem incomplete, I wonder if that says something on the subject.

P.S
so far I only read P&S, Srednicki (completed), Ticciati's (the first couple of chapters) it doesn't seem like QFT is a mathematical consistent theory, too many topics to join them all together...

 

[bhobba]

just have a feeling that something is wrong with this framework of QFT if there are so many stuff that is wrong in the derivations in different books.

Nah - that's not it.

Its as one of my favorite posters says - Strangerep - you find humility in Field Theory - its HARD and the manipulations mind numbing. For guys like me that have trouble doing long calculations without error QFT is your nemesis.

My QFT isn't good enough right now to undertake it, maybe sometime later, but Wienbergs texts by reputation are supposed to be spot on as well as being one of the classics on the subject.

Thanks
Bill

 

[MathematicalPhysicist]

Nah - that's not it.

Its as one of my favorite posters says - Strangerep - you find humility in Field Theory - its HARD and the manipulations mind numbing. For guys like me that have trouble doing long calculations without error QFT is your nemesis.

My QFT isn't good enough right now to undertake it, maybe sometime later, but Wienbergs texts by reputation are supposed to be spot on as well as being one of the classics on the subject.

Thanks
Bill

Weinberg has his share of "problems" with his books.

I'll tell you why I think something is fishy with QFT, I mean if it were legitimate enterprise I am sure there would be one lengthy textbook without errors with all the calculations in it (I was thinking along the lines of Warren Siegel's free ebook), the same feeling I have with Statistical Mechanics.

Sure the calculations are long and tiresome, but someone got them right, right?

Or it's all done by computers nowadays that we don't know how to do the calculations by hand anymore... :-D

 

[vanhees71]

so far I only read P&S, Srednicki (completed), Ticciati's (the first couple of chapters) it doesn't seem like QFT is a mathematical consistent theory, too many topics to join them all together...

Well P&S is a great book, but it has too many typos and sometimes even unacceptable glitches (like dimensionful arguments in logarithms in the chapter about the renormalization group, which imho must not happen, particularly in this context). With Srednicky I have the quibble that he discusses an unstable theory at length (namely $\phi^3$ theory) without ever note the problems with it. I don't know Ticciati. The 3 volumes of Weinberg are exceptional. It's the best textbook series written on the subject. The only problem is that it is not for introductory studies but goes into all the subtle details from the very beginning. That's why I recommended the book by Schwartz.

That said, it's indeed true that relativistic QFT is not a mathematically consistent theory, but renormalized perturbation theory is!

 

[MathematicalPhysicist]

Well P&S is a great book, but it has too many typos and sometimes even unacceptable glitches (like dimensionful arguments in logarithms in the chapter about the renormalization group, which imho must not happen, particularly in this context). With Srednicky I have the quibble that he discusses an unstable theory at length (namely $\phi^3$ theory) without ever note the problems with it. I don't know Ticciati. The 3 volumes of Weinberg are exceptional. It's the best textbook series written on the subject. The only problem is that it is not for introductory studies but goes into all the subtle details from the very beginning. That's why I recommended the book by Schwartz.

That said, it's indeed true that relativistic QFT is not a mathematically consistent theory, but renormalized perturbation theory is!

Well, you cannot prove that it's consistent from within the theory by one of Godel's incompleteness theorems; but if a theory is inconsistent you can prove it by deriving P&~P from its axioms and theorems.

 

[stevendaryl]

Well P&S is a great book, but it has too many typos and sometimes even unacceptable glitches (like dimensionful arguments in logarithms in the chapter about the renormalization group, which imho must not happen, particularly in this context). With Srednicky I have the quibble that he discusses an unstable theory at length (namely $\phi^3$ theory) without ever note the problems with it.

I think $\phi^3$ field theory is just for practice in learning to compute terms in perturbation theory.

 

[atyy]

With Srednicky I have the quibble that he discusses an unstable theory at length (namely $\phi^3$ theory) without ever note the problems with it.

Srednicki notes the problems with $\phi^3$ theory when he introduces it (first page of Chapter 9).

 

[king vitamin]

The problem is simply that a "complete" QFT textbook needs to be thousands of pages, and a thousand page textbook cannot be free from errors. The situation gets better if a book has gone through many editions and/or has an errata (e.g. I think I've only caught one or two errors in Zinn-Justin's tome). Certainly a recent book like Schwartz still has many uncorrected errors (as I am learning right now while grading a QFT course that uses his textbook).

 

[Vanadium 50]

I'll tell you why I think something is fishy with QFT, I mean if it were legitimate enterprise I am sure there would be one lengthy textbook without errors with all the calculations in it

By that same argument, Calculus is not a legitimate enterprise.

 

[bhobba]

Textbooks will always have errors - it's inevitable.

I just recently spotted an error in the mathematics of a post I made years ago and refer to often (it's the proof of Gleason's Theorem) and I was really careful when I posted it. I suppose I should correct it, but gee this is graduate level material. I think students at that level can do a bit of 'tidying up' so to speak themselves so I decided to leave it alone and see if anyone else spots it .

I think it was the author of QFT In a Nutshell that mentioned it - what's the difference between a good theoretical physicist and a bad one - bad ones make errors that give the wrong result - good one make errors that cancel out - or something like that anyway.

Thanks
Bill

 

[MathematicalPhysicist]

I am reading the book called: "Gauge theory of elementary particle physics" by Ta-Pei Cheng and Ling-Fong Li.

On pages 19-20 they write: $(1.78) W[J] = \bigg[\exp(\int d^4x (\mathcal{L}_1 (\frac{\delta}{\delta J}))\bigg] W_0[J]$, where: $W_0[J]=\int [d\phi] \exp\bigg[ \int d^4x (\mathcal{L}_0+J\phi)\bigg]$.

Now, on page 20 they write:" The perturbative expansion in powers of $\mathcal{L}_1$ of the exponential in (1.78) gives:

$$(1.85) W[J] = W_0[J]\bigg\{ 1+\lambda\omega_1[J]+\lambda^2 \omega_2[J]+\ldots \bigg \},$$

where $$(1.86) \omega_1[J] = -\frac{1}{4!}W_0^{-1}[J]\bigg\{ \int d^4x \bigg[\frac{\delta}{\delta J(x)} \bigg]^4 \bigg\} W_0[J]$$

$$\omega_2[J]=-\frac{1}{2(4!)^2} W_0^{-1}[J]\bigg\{ \int d^4x \bigg[ \frac{\delta}{\delta J(x)}\bigg]^4\bigg\}^2 W_0[J] = $$

$$ = -\frac{1}{2(4!)} W_0^{-1}[J]\bigg\{ \int d^4x \bigg[\frac{\delta}{\delta J(x)}\bigg]^4\bigg\} \omega_1[J]$$
Now, for my question, after I plug $\omega_1[J]$ into the above last equation I get:

$$\frac{1}{2(4!)^2} W_0^{-1}[J]\{ \int d^4 x \bigg[ \frac{\delta}{\delta J(x)} \bigg]^4 \} W_0^{-1}[J] \{ \int d^4 x \bigg[ \frac{\delta}{\delta J(x)} \bigg]^4 \} W_0[J]$$

The last expression is not the same as the above expression, i.e. of $-\frac{1}{2(4!)^2} W_0^{-1}[J]\bigg\{ \int d^4x \bigg[ \frac{\delta}{\delta J(x)}\bigg]^4\bigg\}^2 W_0[J]$.

Perhaps instead of $\omega_1[J]$ it should be $-W_0[J] \omega_1[J]$ in equation (1.86)?
I am puzzled, what do you think?

@vanhees71 or anyone else who can help.

After equations (1.85) and (1.86);

When we plug in the explicit form (eqn (1.82)) for $W_0[J]$, we obtain:

$$(1.87) \ \ \ \ \omega_1[J]=-\frac{1}{4!}\bigg[\Delta(x,y_1)\Delta(x,y_2)\Delta(x,y_3)\Delta(x,y_4)J(y_1)J(y_2)J(y_3)J(y_4)+3!\Delta(x,y_1)\Delta(x,y_2)\Delta(x,x)J(y_1)J(y_2)\bigg]$$

and

$$\omega_2[J] = 1/2 \omega_1^2[J]+\frac{1}{2(3!)^2}\Delta(x_1,y_1)\Delta(x_1,y_2)\Delta(x_1,y_3)\Delta(x_1,x_2)\Delta(x_2,y_4)$$
$$\times \Delta(x_2,y_5)\Delta(x_2,y_6)J(y_1)J(y_2)J(y_3)J(y_4)J(y_5)J(y_6)$$
$$+\frac{3}{2(4!)}\Delta(x_1,y_1)\Delta(x_1,y_2)\Delta^2(x_1,x_2)\Delta(x_2,y_3)\Delta(x_2,y_4)$$
$$\times J(y_1)J(y_2)J(y_3)J(y_4)+\frac{2}{2(4!)}\Delta(x_1,y_1)\Delta(x_1,x_1)\Delta(x_1,x_2)$$
$$\times \Delta(x_2,y_2)\Delta(x_2,y_3)\Delta(x_2,y_4)J(y_1)J(y_2)J(y_3)J(y_4)$$
$$+\frac{1}{8}\Delta(x_1,y_1)\Delta(x_1,x_1)\Delta(x_1,x_2)\Delta(x_2,x_2)\Delta(x_2,y_2)J(y_1)J(y_2)$$
$$+\frac{1}{8}\Delta(x_1,y_1)\Delta^2(x_1,x_2)\Delta(x_2,x_2)\Delta(x_1,y_2)J(y_1)J(y_2)$$
$$+\frac{1}{12} \Delta(x_1,y_1)\Delta^3(x_1,x_2)\Delta(x_2,y_2)J(y_1)J(y_2) \ \ \ \ (1.88)$$

where we have dropped all J-independent terms. It ias understood that all arguments $(x_i,y_i)$ are integrated over.

equation (1.82) is: $(1.82) \ \ \ \ W_0[J]=\exp\bigg[ 1/2 \int d^4x d^4y J(x) \Delta(x,y)J(y)$, where $(1.84) \ \ \ \Delta(x,y) = \int\frac{d^4\kappa/((2\pi)^4)}{\kappa^2+\mu^2}e^{i\kappa\cdot (x-y)}$, where $\kappa = (ik_0 , \vec{k})$.

To tell you the truth I don't see how to derive (1.87) and (1.88)?

I mean I don't see how to derive these identities, obviously I only need to take derivatives but where the did the indices of $x_i$ and $y_i$ come from?

Does someone know where this explicit calculation is done?

 

[MathematicalPhysicist]

Can anyone help me with my last post in this thread?

 

[Demystifier]

what's the difference between a good theoretical physicist and a bad one - bad ones make errors that give the wrong result - good one make errors that cancel out - or something like that anyway.

Good ones know the result before the actual calculation, so the error in the calculation does not affect it.

 

[MathematicalPhysicist]

Can someone help me with post #18 in this thread?

Thanks!