Lagrangian "Mass Terms" [Archive]

Yi-Zen Chu; Yiren Qu

Jun7-04, 04:54 AM

Chris Dams

Jun7-04, 12:37 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Dear Yi-Zen,\n\nLet me start by stating that the answers below are given with the\npath integral formulation of quantum field theory in the back of\nmy mind. The answers will not be the same if one has the operator\nformalism in the back of ones mind, but in my opinion it is best\nnot to confuse someone who does not know very much about the subject\nwith these different points of view. These remarks hope to achieve\nthat there will not be some other physicist starting to argue that\nsome of my remarks are untrue in the operator formalism. Constructive\ncritisism is of course welcomed.\n\n"Yi-Zen Chu; Yiren Qu" <y#i#-#z#e#n#.#c#h#u#@#y#a#l#e#.#e#d#u> writes:\n\n>1) What is the meaning of the Lagrangian in QFT? I suppose we plug it\n>into the Euler-Lagrange equations? But what does it give us? Of what\n>object is the Lagrangian?\n\nAs in classical physics, the Lagrangian is supposed to be enough to\nspecify what kind of system we have. If you plug the Lagrangian into\nthe Euler-Lagrange equations, you will find the field equations of\nthe classical (non-quantum verion) of the field theory. Of course, there\nis quite a lot more to the quantum version of the theory. The basic\nobject that one calculates in high energy physics are correlations\n(in QFT these are actually related to transition amplitudes) between\n2 or more fields.\n\n>2) What exactly does the mass term of the Lagrangian mean?\n\nIf one looks at the transition amplitudes, mentioned in 1) these obey a\nwave equation. This can be used to determine, if one starts out with\nsome wave packet, with what velocity the spot where one most likely\nexpects to find the particle moves. If there is no mass term this will\nbe the speed of light. Generally, the square of the four momentum of\na particle that can travel over arbitrary distances is equal to its\nmass squared.\n\n>3) What does the four-component column psi mean? Is it a solution of the\n>Dirac equation? Does (psi_bar gamma_mu psi) correspond to the\n>probability current like in the case for the Dirac equation?\n\nBecause we are dealing with a quantum theory, psi does not obey the\nequations of motion. One could expect that if one takes some kind\nof classical limit (processes where the action is much larger than\nhbar) the classical solutions are the most important contribution.\n\n>4a) Is psi in the Lagrangian the wavefunction of some particle (e.g. the\n>tau neutrino)? Or is the wavefunction of a family of associated\n>particles like that of all neutrinos (tau, mu, electron)? Or is this\n>something we put into the Lagrangian by hand - that is do we multiply\n>the psi\'s of the mu, tau and electron neutrino in a way that is\n>consistent with relativity, left- and right-handedness, hermicity,\n>choice of whether we want our antiparticles to be equal to our\n>particles, etc., take various linear combinations and then see what\n>comes out of it after diagonalization?\n\npsi is not the wave function. A wave function should represent the\nquantum state. In QFT one can, however have a superposition of a\nstate where psi is a certain function with a state where psi is a\ntotally different function. psi is an observable, although it is not\nterribly observable.\n\n>4b) How exactly does one decide what kind of products occur in these\n>mass terms? I quoted things like relativity, antiparticle and particle\n>requirements, handedness, etc. but in reality I don\'t fully understand\n>how these things are exactly implemented as well. Can someone tell me\n>the full list of conditions one needs to consider, and how to go about\n>constructing a general mass term from scratch?\n\nMass terms are polynomials in degree two of the basic fields.\nRelativity is implemented by having a Lagrangian that is covariant, in\nthe sense of special relativity. Particles come about if one considers\ncorrelators between a field at some point in space time and some other\npoint in space time. In case that the momentum of a wave packet is on\nthe mass shell, the wave packet can travel, almost unchanged, over\narbitrary distances. This is interpreted as a particle. If a field\ncontains sufficient degrees of freedom for two particles, these may\nbe called particle and anti-paritcle if they can annihilate to two\nphotons. Most of the time this happens when the field is complex and\ntherefore contains twice the number of degrees of freedom as a real\nfield.\n\n>5) How does the mass term have anything to do with interactions? In my\n>case I\'m guessing that the reason why these various "mass terms" are\n>products of psi_left and psi_right is because the weak interaction\n>involves (1 - gamma_5) psi?\n\nThe mass term of the Lagrangian from which one can derive the Dirac\nequation, is m bar(psi) psi . This happens to be equal to\nm bar(psi_R) psi_L + m bar(psi_L) psi_R . The requirement that the\nLagrangian must be covariant, rather severly limits the possibilities\nhere. I think it may even be true that if one has both neutrino\'s and\nanti-neutrino\'s, it is not at all possible to hava a quadratic term\nin the lagrangian that multiplies psi_R with psi_R, but I am not\nentirely certain of this.\n\n>6) As one might observe my above questions reflects my absolute\n>cluelessness to all these stuff. Where can I find a review where there\n>is some basic explanation of these issues? I need a source that\n>describes the relevant field theoretic concepts so that I can understand\n>this whole business of the Lagrangian mass term.\n\nPerhaps you could have a look at http://www.nikhef.nl/~t58/Monschau.ps.gz .\nThis introduction is rather basic.\n\nBest,\nChris Dams\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Dear Yi-Zen,

Let me start by stating that the answers below are given with the
path integral formulation of quantum field theory in the back of
my mind. The answers will not be the same if one has the operator
formalism in the back of ones mind, but in my opinion it is best
not to confuse someone who does not know very much about the subject
with these different points of view. These remarks hope to achieve
that there will not be some other physicist starting to argue that
some of my remarks are untrue in the operator formalism. Constructive
critisism is of course welcomed.

"Yi-Zen Chu; Yiren Qu" <y#i#-#z#e#n#.#c#h#u#@#y#a#l#e#.#e#d#u> writes:

>1) What is the meaning of the Lagrangian in QFT? I suppose we plug it
>into the Euler-Lagrange equations? But what does it give us? Of what
>object is the Lagrangian?

As in classical physics, the Lagrangian is supposed to be enough to
specify what kind of system we have. If you plug the Lagrangian into
the Euler-Lagrange equations, you will find the field equations of
the classical (non-quantum verion) of the field theory. Of course, there
is quite a lot more to the quantum version of the theory. The basic
object that one calculates in high energy physics are correlations
(in QFT these are actually related to transition amplitudes) between
2 or more fields.

>2) What exactly does the mass term of the Lagrangian mean?

If one looks at the transition amplitudes, mentioned in 1) these obey a
wave equation. This can be used to determine, if one starts out with
some wave packet, with what velocity the spot where one most likely
expects to find the particle moves. If there is no mass term this will
be the speed of light. Generally, the square of the four momentum of
a particle that can travel over arbitrary distances is equal to its
mass squared.

>3) What does the four-component column \psi mean? Is it a solution of the
>Dirac equation? Does (\psi_bar \gamma_mu \psi) correspond to the
>probability current like in the case for the Dirac equation?

Because we are dealing with a quantum theory, \psi does not obey the
equations of motion. One could expect that if one takes some kind
of classical limit (processes where the action is much larger than
\hbar) the classical solutions are the most important contribution.

>4a) Is \psi in the Lagrangian the wavefunction of some particle (e.g. the
>\tau neutrino)? Or is the wavefunction of a family of associated
>particles like that of all neutrinos (\tau, \mu, electron)? Or is this
>something we put into the Lagrangian by hand - that is do we multiply
>the \psi's of the \mu, \tau and electron neutrino in a way that is
>consistent with relativity, left- and right-handedness, hermicity,
>choice of whether we want our antiparticles to be equal to our
>particles, etc., take various linear combinations and then see what
>comes out of it after diagonalization?

\psi is not the wave function. A wave function should represent the
quantum state. In QFT one can, however have a superposition of a
state where \psi is a certain function with a state where \psi is a
totally different function. \psi is an observable, although it is not
terribly observable.

>4b) How exactly does one decide what kind of products occur in these
>mass terms? I quoted things like relativity, antiparticle and particle
>requirements, handedness, etc. but in reality I don't fully understand
>how these things are exactly implemented as well. Can someone tell me
>the full list of conditions one needs to consider, and how to go about
>constructing a general mass term from scratch?

Mass terms are polynomials in degree two of the basic fields.
Relativity is implemented by having a Lagrangian that is covariant, in
the sense of special relativity. Particles come about if one considers
correlators between a field at some point in space time and some other
point in space time. In case that the momentum of a wave packet is on
the mass shell, the wave packet can travel, almost unchanged, over
arbitrary distances. This is interpreted as a particle. If a field
contains sufficient degrees of freedom for two particles, these may
be called particle and anti-paritcle if they can annihilate to two
photons. Most of the time this happens when the field is complex and
therefore contains twice the number of degrees of freedom as a real
field.

>5) How does the mass term have anything to do with interactions? In my
>case I'm guessing that the reason why these various "mass terms" are
>products of \psi_left and \psi_right is because the weak interaction
>involves (1 - \gamma_5) \psi?

The mass term of the Lagrangian from which one can derive the Dirac
equation, is m bar(\psi) \psi . This happens to be equal to
m bar(\psi_R) \psi_L + m bar(\psi_L) \psi_R . The requirement that the
Lagrangian must be covariant, rather severly limits the possibilities
here. I think it may even be true that if one has both neutrino's and
anti-neutrino's, it is not at all possible to hava a quadratic term
in the lagrangian that multiplies \psi_R with \psi_R, but I am not
entirely certain of this.

>6) As one might observe my above questions reflects my absolute
>cluelessness to all these stuff. Where can I find a review where there
>is some basic explanation of these issues? I need a source that
>describes the relevant field theoretic concepts so that I can understand
>this whole business of the Lagrangian mass term.

Perhaps you could have a look at http://www.nikhef.nl/~t58/Monschau.ps.gz .
This introduction is rather basic.

Best,
Chris Dams