Questions on An Introduction to Quantum Field Theory by Peskin and Schroeder

[fliptomato]

Greetings--I have a few questions from An Introduction to Quantum Field Theory by Peskin and Schroeder.

Note: I'm not sure how to construct the contraction symbol using [tex]\LaTeX[/tex], so instead I will use the following cumbersome convention: [tex]\overbrace{\psi(x)\overline{\psi(y)}}=S_F(x-y)[/tex], they feynman propagator for a spin 1/2 particle, as in equation (4.111) of P&S.

First of all, I'm a little skeptical about the product [tex]\overline{\psi(y)}\psi(x)[/tex] where [tex]\psi(y) \equiv \psi(y)^\dag \gamma^0 [/tex] because the order seems backward. The product [tex]\overline{\psi}\psi[/tex] is ok because it is the "conventional" matrix multiplication of a row vector with a column vector to yield a scalar (with a gamma matrix inside). However, [tex]\psi \overline{\psi}[/tex] is the multriplication of a column vector by a [row vector times a gamma matrix]. This doesn't seem to make sense, and hence the commutators (say, the bottom of p. 54) with [tex]\overline{\psi}[/tex] and [tex]\psi[/tex] don't seem well defined to me. Thus, similarly, I'm unhappy with the definition of the contraction [tex]\overbrace{\psi(x)\overline{\psi(y)}}[/tex] in (4.108) on p. 116.

Moving on with the fermion Feynman rules, on p.119 P&S say "note that in our examples the Dirac indices contract together along fermion lines." I imagine this has to do with the order in which we translate Feynman diagrams into amplitudes (i.e. against the fermion arrows). However I'm not sure how this rule is "derived." Is this because one needs to move around fermion and boson operators (using anticommutation and commutation relations) such that the contractions are "nested": [tex]\overbrace{\psi\overbrace{\psi\overline{\psi}}\overline{\psi}}[/tex]?

Related to my uncomfortability with these contractions is the closed fermion loop that P&S describe at the bottom of p.120. I'm unclear about the derivation of equation (4.120) in which a closed loop of fermions yields a trace of fermion propagators. ((I'm happy with where the (-1) came from, the anticommutation relations of the [tex]\psi[/tex]s)). How does one go from the series of contractions [tex]\overbrace{\overline{\psi}\overbrace{\psi\overline{\psi}}\overbrace{\psi\overline{\psi}}\overbrace{\psi\overline{\psi}}\psi}[/tex] to a trace of these contractions? It seems like I'm not really understanding these fermion contractions.

Any assistance would be much appreciated!

Thanks,
Flip
flipt *at* stanford *dot* edu

 

[dextercioby]

Greetings--I have a few questions from An Introduction to Quantum Field Theory by Peskin and Schroeder.

Note: I'm not sure how to construct the contraction symbol using [tex]\LaTeX[/tex], so instead I will use the following cumbersome convention: [tex]\overbrace{\psi(x)\overline{\psi(y)}}=S_F(x-y)[/tex], they feynman propagator for a spin 1/2 particle, as in equation (4.111) of P&S.

Well,let's see if i can:
[tex] S_{F}(x-y)=:\overline{\Psi(x)\overline{\Psi(y)}} [/tex]
Apparently,i did... Though i think "overbrace" will do for terms inbetween the contracted terms.

First of all, I'm a little skeptical about the product [tex]\overline{\psi(y)}\psi(x)[/tex] where [tex]\psi(y) \equiv \psi(y)^\dag \gamma^0 [/tex] because the order seems backward. The product [tex]\overline{\psi}\psi[/tex] is ok because it is the "conventional" matrix multiplication of a row vector with a column vector to yield a scalar (with a gamma matrix inside). However, [tex]\psi \overline{\psi}[/tex] is the multriplication of a column vector by a [row vector times a gamma matrix]. This doesn't seem to make sense, and hence the commutators (say, the bottom of p. 54) with [tex]\overline{\psi}[/tex] and [tex]\psi[/tex] don't seem well defined to me.

You're right.It doesn't make sense.However,there is no worry about such problems.Remember where such terms (spinor fields in 'arbitrary' order) appear.In the S-matrix when evaluating the Dyson chronological operator on the product of interaction Hamiltonians.Expanding it according to Wick's theorem involves such awkward arrangements of spinor fields,but they usually come in pairs and with a gamma matrix in between.Think about the QED interaction hamiltonian and apply Wick's theorem for the second order term:

[tex] \hat{T}\{\hat{N}[\overline{\Psi}_{\alpha}(x)(\gamma_{\mu})^{\alpha}_{\beta}\Psi^{\beta}(x)A^{\mu}(x)]\hat{N}[\overline{\Psi}_{\alpha'}(y)(\gamma_{\nu})^{\alpha'}_{\beta'}\Psi^{\beta'}(y)A^{\nu}(y)]\} [/tex]

[tex]=\hat{T}\{\hat{N}[\overline{\Psi}_{\alpha}(x)(\gamma_{\mu})^{\alpha}_{\beta}\Psi^{\beta}(x)]\hat{N}[\overline{\Psi}_{\alpha'}(y)(\gamma_{\nu})^{\alpha'}_{\beta'}\Psi^{\beta'}(y)]\}\hat{T}[A^{\mu}(x)A^{\nu}(y)] [/tex]

[tex]=\{\hat{N}[\overline{\Psi}_{\alpha}(x)(\gamma_{\mu})^{\alpha}_{\beta}\Psi^{\beta}(x)\overline{\Psi}_{\alpha'}(y)(\gamma_{\nu})^{\alpha'}_{\beta'}\Psi^{\beta'}(y)]+\hat{N}[\overbrace{\overline{\Psi}_{\alpha}(x)(\gamma_{\mu})^{\alpha}_{\beta}\Psi^{\beta}(x)\overline{\Psi}_{\alpha'}(y)(\gamma_{\nu})^{\alpha'}_{\beta'}\Psi^{\beta'}(y)}][/tex]

[tex]+\hat{N}[\overline{\Psi}_{\alpha}(x)(\gamma_{\mu})^{\alpha}_{\beta}\overbrace{\Psi^{\beta}(x)\overline{\Psi}_{\alpha'}(y)}(\gamma_{\nu})^{\alpha'}_{\beta'}\Psi^{\beta'}(y)]+\hat{N}[\overbrace{\overline{\Psi}_{\alpha}(x)(\gamma_{\mu})^{\alpha}_{\beta}\overbrace{\Psi^{\beta}(x)\overline{\Psi}_{\alpha'}(y)}(\gamma_{\nu})^{\alpha'}_{\beta'}\Psi^{\beta'}(y)}]\}\hat{T}[A^{\mu}(x)A^{\nu}(y)] [/tex]

Look at these contractions very wel,at the order of the fields and at matrix multiplication rule.The chronological contraction is not a multiplication of matrices.It is simply an operator which acts on 2 fields (fermionic in this case) and outputs a complex function,namely the Feynman propagator.To this operator,it doens't matter the order of the 2 fields wrt to matrix multiplication.It matters only that they are fermionic,so changing the order in the chronologically contracted product of 2 fermionic fields produces a sign minus.The matrix multiplication rule is unaffected,since the fields keep their initial position wrt to each other and to the gamma matrices.In fact,the result of a chronological product between 2 fields is the propagator which is itself a matrix,but being a Grassmann zero parity variable with indices summed over,it can move both left and right in such a complicated s***,i.e.a sum over normal products.
I tried to put spinor indices at the gamma matrices according to the rule NW-SE,but i couldn't ...Sorry...


Let's take the last normal product:
[tex] \hat{N}[\overbrace{\overline{\Psi}_{\alpha}(x)(\gamma_{\mu})^{\alpha}_{\beta}\overbrace{\Psi^{\beta}(x)\overline{\Psi}_{\alpha'}(y)}(\gamma_{\nu})^{\alpha'}_{\beta'}\Psi^{\beta'}(y)}][/tex]

[tex]=[iS_{F}(x-y)]^{\beta'}_{\alpha}(\gamma_{\mu})^{\alpha}_{\beta}[iS_{F}(y-x)]^{\beta}_{\alpha'}(\gamma_{\nu})^{\alpha'}_{\beta'}=Trace[iS_{F}(x-y)\gamma_{\mu}iS_{F}(y-x)\gamma_{\nu}][/tex]

Again,all spinor indices are summed over in the NW-SE convention.The trace appears naturally,since all indices are being summed over.Now u can see very simply that the term above indicates a fermionic loop.Coupled with the 2 photonic propagators and integrated over x and y,plus the factor 1/2 u get the S matrix element for the eigenenergy of the fermion.


Daniel.

 

[dextercioby]

Moving on with the fermion Feynman rules, on p.119 P&S say "note that in our examples the Dirac indices contract together along fermion lines." I imagine this has to do with the order in which we translate Feynman diagrams into amplitudes (i.e. against the fermion arrows).

Nope,it's got to do with evaluating S-matrix elements and using the Feynman rules for the propagators.Those rules include spinor indices (for the fermionic case,actually "mu"Lorentz index can be thought as a pair of spinor indices,one dotted and one not) and they simply get summed over when computing the 'amplitude' M_{fi} according to the Feynman diagram.

However I'm not sure how this rule is "derived." Is this because one needs to move around fermion and boson operators (using anticommutation and commutation relations) such that the contractions are "nested": [tex]\overbrace{\psi\overbrace{\psi\overline{\psi}}\overline{\psi}}[/tex]?

No,i told u,it's simple diagramatics.That "moving" induces the diagrams and not viceversa.Things would be much more clear if u let aside this horrible operator approach and deal with this issue using the Path-integral approach.When computing the Green functions,ther's no ordering to take into account,as all propagators are bosonic (zero Grassmann parity) and they can be moved around without any problem.The "sources" are fermionic variables and that spoil the fun,but hey,it can't be that simple,right??If it was,it could be taught in kindergarten,right??

Related to my uncomfortability with these contractions is the closed fermion loop that P&S describe at the bottom of p.120. I'm unclear about the derivation of equation (4.120) in which a closed loop of fermions yields a trace of fermion propagators. ((I'm happy with where the (-1) came from, the anticommutation relations of the [tex]\psi[/tex]s)). How does one go from the series of contractions [tex]\overbrace{\overline{\psi}\overbrace{\psi\overline{\psi}}\overbrace{\psi\overline{\psi}}\overbrace{\psi\overline{\psi}}\psi}[/tex] to a trace of these contractions? It seems like I'm not really understanding these fermion contractions.

It's not a big deal,really,just care with anticommuting Grassmann parity 1 operators.I'm sure u know how to apply Wick's theorem for various scattering processes.As i told u already,if u painted down all spinor indices,u'd be able to see those traces,else,no.

Daniel.

 

[fliptomato]

Thanks very much--that was illuminating! Your treatment leads me to ask if it would be fair to think of the contraction as a kind of "outer product" in which a column vector is 'dotted' into a row vector to produce a matrix (the Feynman propagator).

Also, I am not familiar with Grassman numbers, though they seem to pop up every now and then. Is there a suggested reference for introductory reading (w/rt QFT)?

Thanks,
Flip

 

[marlon]

Thanks very much--that was illuminating! Your treatment leads me to ask if it would be fair to think of the contraction as a kind of "outer product" in which a column vector is 'dotted' into a row vector to produce a matrix (the Feynman propagator).

Also, I am not familiar with Grassman numbers, though they seem to pop up every now and then. Is there a suggested reference for introductory reading (w/rt QFT)?

Thanks,
Flip

basically Grassmann variables are defined as anti-commuting variables and in QFT they represent the fermions. The reason why fermions (or at least the variables describing them) are anti-commutative lies in the fact that they need to obey the spin-statistic-theorem...

Go to my journal and read the "info on the web"entry. There is a link to a great free online course on QFT...

regards
marlon

 

[dextercioby]

Also, I am not familiar with Grassman numbers, though they seem to pop up every now and then. Is there a suggested reference for introductory reading (w/rt QFT)?

Thanks,
Flip

Here's the deal.You said u were working after Paskin & Schroeder.Doesn't have a chapter on fermionic variables??The book by Bailin & Love (David Bailin & Alexander Love:"Introduction to gauge field theory",Bristol,Adam Hilger,Second Edition 1993) has a chapter on fermionic/Grassmann variables.It's pretty readable.
If u had taken a course on BRST quantization,u would have certainly be taught Grassmann/supersymmetric/Z_{2} graded algebras.
I've been introduced to Grassmann variables in the second year of school,in the course on Analytic Mechanics.

Daniel.