Covariant Derivative question [Archive]

Flip Tomato

Aug14-04, 06:58 AM

<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>\n\nGreetings--I\'m trying to do some reading into relativistic quantum mechanics\n(I\'ve just taken quantum at the undergrad level) and I\'m curious about the\ncovariant derivative that is used when discussing gauge invariance.\n\nWhat motivates the definition of the gauge covariant derivative, other than\nthat it gives a nice result? I understand the gradient (1) mathematically as\na derivative operator and (2) physically as a momentum operator--but the\ngauge covariant derivative doesn\'t have any intuitive appeal to me other\nthan sketchily looking like it would involve the gauge freedom of choosing\nA.\n\nIn Chris Quigg\'s "Gauge Theories of the..." book, he writes: "Local phase\ninvariance may be achieved if the equations of motion and the observables\ninvolving derivatives are modified by the introduction of the\nelectromagnetic field A_\\mu(x). If the gradient is everywhere replaced by\nthe gauge covariant derivative, [this is satisfied]."\n\nIn terms of the big picture, I understand (but please correct me if i\'m\nwrong) that local phase invariance is a general principle that we would like\nto have in quantum mechanics, so we impose it by introducing this gauge\ncovariant derivative. The term in the gauge covariatn derivative, A_\\mu(x),\nthen *turns out* to be the EM potential and lo and behold, E&M pops out of\nthis principle of local phase invariance. The fact that E&M naturally pops\nout of this principle--this is "evidence" to believe that local phase\ninvariance is a reasonable "first principle"?\n\nI know my questions are a little hazy right now as I\'m still trying to get\nmy head around these topics--but any insight would be much appreciated (and\nprobably followed by more precise questions).\n\nThanks,\nFlip Tanedo\nflipt (at) stanford (dot) edu\n\nPS--on a totally unrelated note, I\'m not very good with literature searches\nyet... how do I find *review* articles in a subject that I\'m interested in\nstudying?\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>Greetings--I'm trying to do some reading into relativistic quantum mechanics
(I've just taken quantum at the undergrad level) and I'm curious about the
covariant derivative that is used when discussing gauge invariance.

What motivates the definition of the gauge covariant derivative, other than
that it gives a nice result? I understand the gradient (1) mathematically as
a derivative operator and (2) physically as a momentum operator--but the
gauge covariant derivative doesn't have any intuitive appeal to me other
than sketchily looking like it would involve the gauge freedom of choosing
A.

In Chris Quigg's "Gauge Theories of the..." book, he writes: "Local phase
invariance may be achieved if the equations of motion and the observables
involving derivatives are modified by the introduction of the
electromagnetic field A_\mu(x). If the gradient is everywhere replaced by
the gauge covariant derivative, [this is satisfied]."

In terms of the big picture, I understand (but please correct me if i'm
wrong) that local phase invariance is a general principle that we would like
to have in quantum mechanics, so we impose it by introducing this gauge
covariant derivative. The term in the gauge covariatn derivative, A_\mu(x),
then *turns out* to be the EM potential and lo and behold, E&M pops out of
this principle of local phase invariance. The fact that E&M naturally pops
out of this principle--this is "evidence" to believe that local phase
invariance is a reasonable "first principle"?

I know my questions are a little hazy right now as I'm still trying to get
my head around these topics--but any insight would be much appreciated (and
probably followed by more precise questions).

Thanks,
Flip Tanedo
flipt (at) stanford (dot) edu

PS--on a totally unrelated note, I'm not very good with literature searches
yet... how do I find *review* articles in a subject that I'm interested in
studying?

Hendrik van Hees

Aug14-04, 11:08 AM

<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>\nFlip Tomato wrote:\n\n> Greetings--I\'m trying to do some reading into relativistic quantum\n> mechanics (I\'ve just taken quantum at the undergrad level) and I\'m\n> curious about the covariant derivative that is used when discussing\n> gauge invariance.\n>\n> What motivates the definition of the gauge covariant derivative, other\n> than that it gives a nice result? I understand the gradient (1)\n> mathematically as a derivative operator and (2) physically as a\n> momentum operator--but the gauge covariant derivative doesn\'t have any\n> intuitive appeal to me other than sketchily looking like it would\n> involve the gauge freedom of choosing A.\n>\n> In Chris Quigg\'s "Gauge Theories of the..." book, he writes: "Local\n> phase invariance may be achieved if the equations of motion and the\n> observables involving derivatives are modified by the introduction of\n> the electromagnetic field A_\\mu(x). If the gradient is everywhere\n> replaced by the gauge covariant derivative, [this is satisfied]."\n>\n> In terms of the big picture, I understand (but please correct me if\n> i\'m wrong) that local phase invariance is a general principle that we\n> would like to have in quantum mechanics, so we impose it by\n> introducing this gauge covariant derivative. The term in the gauge\n> covariatn derivative, A_\\mu(x), then *turns out* to be the EM\n> potential and lo and behold, E&M pops out of this principle of local\n> phase invariance. The fact that E&M naturally pops out of this\n> principle--this is "evidence" to believe that local phase invariance\n> is a reasonable "first principle"?\n>\n> I know my questions are a little hazy right now as I\'m still trying to\n> get my head around these topics--but any insight would be much\n> appreciated (and probably followed by more precise questions).\n\nOne of the most important principles in physics are symmetries.\nEspecially in quantum theory it is needed to make the link between\nobservables and the mathematical formalism. Unfortunately, in many\ntextbooks this point is not made clear from the very beginning, but\nthey use some handwaving arguments like "canonical quantisation". It\'s\na pity that this procedure became the name "canonical". By chance, it\nworks for some cases, for most not. For instance, it goes wrong even in\nrelatively simple cases, when you try to quantise the free\nnon-relativistic spin-0 particle in spherical coordinates.\n\nThe real good quantisation starts from symmetry principles. First of all\nyou like a quantum theory which implements the space-time structure\nunder consideration. In the case of relativistic quantum theory that is\nthe Einstein-Minkowski space time.\n\nThen you have the principles of quantum theory which are independent\nfrom the actual realisation and thus valid for all quantum systems\n(relativistic, non-relativistic, two-level models etc. etc.):\n\n(1) A system is described by states. States are represented by rays in\nHilbert space, or equivalently by a member of the projective Hilbert\nspace, i.e. A ray in Hilbert space is the set [psi], which is defined\nfor a given |psi> \\neq 0 \\in H\n\n[psi]=3D{lambda |psi>|lambda \\in C}\n\nIf the system is prepared to be in state [phi], the probability to find\nit in state [psi] is given by\n\nP_{phi}(psi)=3D|<phi|psi>|^2/(<phi|phi><psi|psi>)\n\nIt is easy to see that this probability and thus all predictions about\nmeasurements fromquantum theory is independent of the choice of\nrepresentatives of the states [phi] and [psi].\n\nIt is important to keep in mind that not the kets |psi> are representing\nthe state of a system but the rays [psi].\n\n(2) Observables are represented by selfadjoint operators. The possible\noutcomes of an exact measurement of an observable is given by the\nspectrum (generalised eigenvalues) of the corresponding operator.\n\nI do not want to go into the very complicated question of how to prepare\nsystems in a certain state through measurements and filtering\nprocesses. You might read the introductory chapter in Sakurai\'s book\n"Modern Quantum Mechanics".\n\nNow comes the more interesting step to fill this concepts with real\nphysics, and as stressed before, for this we need the notion of\nsymmetries. In QT, a symmetry is given by a map of the states (rays in\nHilbert space) and observables (selfadjoint operators in Hilbert space)\nsuch that all outcomes of predictions about experiments are unchanged.\n\nTo implement the space-time structure consistently in quantum theory,\none has to make sure that the symmetry operations on space time are\nsymmetries of the quantum theory, you like to construct.\n\nNow it turns out that there is a little complication from the fact that\nstates are rays and not vectors in Hilbert space, but at the same time\nit also turns out that exactly this is crucial to get the right answer\nabout real systems, consistent with all experiments done so far.\n\nThe important thing is that all symmetry operations together build a\ngroup, and thus one has to look for all ray representations of the\ngroup. Such representations are rather complicated to deal with and\nthus it is important that there is the Wigner-Bargmann theorem:\n\nEach ray representation can be "lifted" to a unitary or antiunitary\ntransformation of the central extension of the universal covering group\nof the symmetry group of the system.\n\nIn our case of the Poincaregroup (inhomogeneous Lorentz group) this\nmeans that we need to find only the unitary irreducible representations\nof the Poincaregroup where the homogeneous Lorentz transformations,\ni.e., the group SO(1,3) is substituted by its covering group SL(2,C). A\nsystem, described by such an unitary irred. representation is called a\nfree elementary particle.\n\nThen it comes out that there are two large classes of such\nrepresentations which lead to a physically interpretable quantum\ntheory.=20\n\n(a) Elementary particles with a finite mass m (m^2>0)\n(b) Massless elementary particles: m^2=3D0\n\nThe former are further determined by the spin of the particles, which\ndetermines how the state kets of particles at rest change under\nrotations. This case is not much more involved than the\nnon-relativistic particles (although the Galilei group, underlying\nnon-relativistic physics is a little bit more complicated, since there\nare non-trivial different central extensions which do not lead to\nphysically meaningful theories as was shown by Wigner and Ion=FC).=20\n\nIn the standard model there are only very view of these representations\nnecessary: Particles with spin 0 (the Higgs boson, which perhaps is a\nmathematical artifact and not a real particle, but let\'s wait what LHC\ntells us about it) and particles with spin 1/2 (Quarks and leptons) in\nthe Dirac representation (perhaps the neutrinos are in fact Majorana\nfermions, but this is also not completely clarified todate).\n\nThe massless case is more involved. There is no spin, because these\nparticles cannot be at rest. Instead you have the helicity, which is a\nspinlike quantity. It can be interpreted as the projection of the spin\nin direction of the momentum of the particle (in an arbitrarily chosen\nstandard direction, mostly chosen as the z-direction). The helicities\ncan be 0, \\pm 1/2, \\pm 1, etc.\n\nFor helicity 0 and \\pm 1/2 there is no more trouble than in the case of\nmassive particles. In a properly chosen convention, these cases can be\ntreated as the limits of the massive cases with the mass taken to 0.\n\nThis is not true for particles with helicities \\pm 1. Here, precisely\nthis limit becomes ambiguous. From group theory, it is clear why: One\ncannot represent a massless helicity-1-particle by a set of wave\nfunctions. There is no function space, which realises this\nrepresentations, but only a quotient space, i.e., massless\nhelicity-1-fields are represented by vector fields modulo pure gauge\nfields, i.e., if the fields A_{\\mu}(x) is a representation, then for\nany scalar fields \\chi also the field=20\n\nA_{\\mu}\'(x)=3DA_{\\mu}(x)+\\partial_ {\\mu} \\chi(x)=20\n\nrepresents the same physics. Thus, from space-time symmetry, it follows\nthat there must be an additional symmetry for massless\nhelicity-1-particles: The gauge symmetry!\n\nNow you like to describe interacting particles. Then this gauge symmetry\nmust stay a symmetry under any circumstances, because otherwise you get\ninconsistent with the strucure of space and time!\n\nTaken together these principles with the demand of renormalisability,\nyou end up with the minimal coupling principle, i.e., you introduce the\ninteractions of massless helicity-one-particles by the substitution of\nderivatives with covariant derivatives.\n\n>From this point of view, you can realise the gauge principle as well\nwith abelian as with nonabelian gauge groups. As the success of the\nstandard model shows, both realisations are important to describe the\nelementary particles.\n\nAs a textbook about all this, I can recommend only one source:\n\nS. Weinberg, The Quantum Theory of Fields, Vol. I+II\n\nVol. III is about the extension of symmetry principles to supersymmetry.\n\nOf course the more theoretical study of Weinberg\'s books should be\nsupplemented by other texts which show in more detail how actual\ncalculations are to be done.\n\nWith a little bad feeling, for this purpose, I suggest\n\nPeskin/Schroeder, An Introduction to Quantum Field theory\n\nto be read with care. The erratum list on the textbook\'s homepage is a\nmandatory source of clarification, although some conceptual mistakes\n(especially in the chapter about the spontaneously broken linear sigma\nmodel) are not solved there.\n\n--=20\nHendrik van Hees Cyclotron Institute=20\nPhone: +1 979/845-1411 Texas A&M University=20\nFax: +1 979/845-1899 Cyclotron Institute, MS-3366\nhttp://theory.gsi.de/~vanhees/ College Station, TX 77843-3366\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>Flip Tomato wrote:

> Greetings--I'm trying to do some reading into relativistic quantum
> mechanics (I've just taken quantum at the undergrad level) and I'm
> curious about the covariant derivative that is used when discussing
> gauge invariance.
>
> What motivates the definition of the gauge covariant derivative, other
> than that it gives a nice result? I understand the gradient (1)
> mathematically as a derivative operator and (2) physically as a
> momentum operator--but the gauge covariant derivative doesn't have any
> intuitive appeal to me other than sketchily looking like it would
> involve the gauge freedom of choosing A.
>
> In Chris Quigg's "Gauge Theories of the..." book, he writes: "Local
> phase invariance may be achieved if the equations of motion and the
> observables involving derivatives are modified by the introduction of
> the electromagnetic field A_\mu(x). If the gradient is everywhere
> replaced by the gauge covariant derivative, [this is satisfied]."
>
> In terms of the big picture, I understand (but please correct me if
> i'm wrong) that local phase invariance is a general principle that we
> would like to have in quantum mechanics, so we impose it by
> introducing this gauge covariant derivative. The term in the gauge
> covariatn derivative, A_\mu(x), then *turns out* to be the EM
> potential and lo and behold, E&M pops out of this principle of local
> phase invariance. The fact that E&M naturally pops out of this
> principle--this is "evidence" to believe that local phase invariance
> is a reasonable "first principle"?
>
> I know my questions are a little hazy right now as I'm still trying to
> get my head around these topics--but any insight would be much
> appreciated (and probably followed by more precise questions).

One of the most important principles in physics are symmetries.
Especially in quantum theory it is needed to make the link between
observables and the mathematical formalism. Unfortunately, in many
textbooks this point is not made clear from the very beginning, but
they use some handwaving arguments like "canonical quantisation". It's
a pity that this procedure became the name "canonical". By chance, it
works for some cases, for most not. For instance, it goes wrong even in
relatively simple cases, when you try to quantise the free
non-relativistic spin-0 particle in spherical coordinates.

The real good quantisation starts from symmetry principles. First of all
you like a quantum theory which implements the space-time structure
under consideration. In the case of relativistic quantum theory that is
the Einstein-Minkowski space time.

Then you have the principles of quantum theory which are independent
from the actual realisation and thus valid for all quantum systems
(relativistic, non-relativistic, two-level models etc. etc.):

(1) A system is described by states. States are represented by rays in
Hilbert space, or equivalently by a member of the projective Hilbert
space, i.e. A ray in Hilbert space is the set [\psi], which is defined
for a given |\psi> \neq\in H[\psi]=3D{\lambda |\psi>|\lambda \in C}

If the system is prepared to be in state [\phi], the probability to find
it in state [\psi] is given by

P_{\phi}(\psi)=3D|<\phi|\psi>|^2/(<\phi|\phi><\psi|\psi>)

It is easy to see that this probability and thus all predictions about
measurements fromquantum theory is independent of the choice of
representatives of the states [\phi] and [\psi].

It is important to keep in mind that not the kets |\psi> are representing
the state of a system but the rays [\psi].

(2) Observables are represented by selfadjoint operators. The possible
outcomes of an exact measurement of an observable is given by the
spectrum (generalised eigenvalues) of the corresponding operator.

I do not want to go into the very complicated question of how to prepare
systems in a certain state through measurements and filtering
processes. You might read the introductory chapter in Sakurai's book
"Modern Quantum Mechanics".

Now comes the more interesting step to fill this concepts with real
physics, and as stressed before, for this we need the notion of
symmetries. In QT, a symmetry is given by a map of the states (rays in
Hilbert space) and observables (selfadjoint operators in Hilbert space)
such that all outcomes of predictions about experiments are unchanged.

To implement the space-time structure consistently in quantum theory,
one has to make sure that the symmetry operations on space time are
symmetries of the quantum theory, you like to construct.

Now it turns out that there is a little complication from the fact that
states are rays and not vectors in Hilbert space, but at the same time
it also turns out that exactly this is crucial to get the right answer
about real systems, consistent with all experiments done so far.

The important thing is that all symmetry operations together build a
group, and thus one has to look for all ray representations of the
group. Such representations are rather complicated to deal with and
thus it is important that there is the Wigner-Bargmann theorem:

Each ray representation can be "lifted" to a unitary or antiunitary
transformation of the central extension of the universal covering group
of the symmetry group of the system.

In our case of the Poincaregroup (inhomogeneous Lorentz group) this
means that we need to find only the unitary irreducible representations
of the Poincaregroup where the homogeneous Lorentz transformations,
i.e., the group SO(1,3) is substituted by its covering group SL(2,C). A
system, described by such an unitary irred. representation is called a
free elementary particle.

Then it comes out that there are two large classes of such
representations which lead to a physically interpretable quantum
theory.=20

(a) Elementary particles with a finite mass m (m^2>0)
(b) Massless elementary particles: m^2=3D0

The former are further determined by the spin of the particles, which
determines how the state kets of particles at rest change under
rotations. This case is not much more involved than the
non-relativistic particles (although the Galilei group, underlying
non-relativistic physics is a little bit more complicated, since there
are non-trivial different central extensions which do not lead to
physically meaningful theories as was shown by Wigner and Ion=FC).=20

In the standard model there are only very view of these representations
necessary: Particles with spin (the Higgs boson, which perhaps is a
mathematical artifact and not a real particle, but let's wait what LHC
tells us about it) and particles with spin 1/2 (Quarks and leptons) in
the Dirac representation (perhaps the neutrinos are in fact Majorana
fermions, but this is also not completely clarified todate).

The massless case is more involved. There is no spin, because these
particles cannot be at rest. Instead you have the helicity, which is a
spinlike quantity. It can be interpreted as the projection of the spin
in direction of the momentum of the particle (in an arbitrarily chosen
standard direction, mostly chosen as the z-direction). The helicities
can be 0, \pm 1/2, \pm 1, etc.

For helicity and \pm 1/2 there is no more trouble than in the case of
massive particles. In a properly chosen convention, these cases can be
treated as the limits of the massive cases with the mass taken to .

This is not true for particles with helicities \pm 1. Here, precisely
this limit becomes ambiguous. From group theory, it is clear why: One
cannot represent a massless helicity-1-particle by a set of wave
functions. There is no function space, which realises this
representations, but only a quotient space, i.e., massless
helicity-1-fields are represented by vector fields modulo pure gauge
fields, i.e., if the fields A_{\mu}(x) is a representation, then for
any scalar fields \chi also the field=20

A_{\mu}'(x)=3DA_{\mu}(x)+\partial_{\mu} \chi(x)=20

represents the same physics. Thus, from space-time symmetry, it follows
that there must be an additional symmetry for massless
helicity-1-particles: The gauge symmetry!

Now you like to describe interacting particles. Then this gauge symmetry
must stay a symmetry under any circumstances, because otherwise you get
inconsistent with the strucure of space and time!

Taken together these principles with the demand of renormalisability,
you end up with the minimal coupling principle, i.e., you introduce the
interactions of massless helicity-one-particles by the substitution of
derivatives with covariant derivatives.

>From this point of view, you can realise the gauge principle as well
with abelian as with nonabelian gauge groups. As the success of the
standard model shows, both realisations are important to describe the
elementary particles.

As a textbook about all this, I can recommend only one source:

S. Weinberg, The Quantum Theory of Fields, Vol. I+II

Vol. III is about the extension of symmetry principles to supersymmetry.

Of course the more theoretical study of Weinberg's books should be
supplemented by other texts which show in more detail how actual
calculations are to be done.

With a little bad feeling, for this purpose, I suggest

Peskin/Schroeder, An Introduction to Quantum Field theory

to be read with care. The erratum list on the textbook's homepage is a
mandatory source of clarification, although some conceptual mistakes
(especially in the chapter about the spontaneously broken linear \sigma
model) are not solved there.

--=20
Hendrik van Hees Cyclotron Institute=20
Phone: +1 979/845-1411 Texas A&M University=20
Fax: +1 979/845-1899 Cyclotron Institute, MS-3366
http://theory.gsi.de/~vanhees/ College Station, TX 77843-3366

Arkadiusz Jadczyk

Aug16-04, 12:55 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>\n\n\nOn 14 Aug 2004 07:58:04 -0400, "Flip Tomato" <flipt@stanford.edu> wrote:\n\n>\n>What motivates the definition of the gauge covariant derivative, other than\n>that it gives a nice result?\n\n\nThere are many possible answers to this question. Let me propose one\nthat appeals to me.\n\nConsider quantum particle in the external EM field. Compute "velocity\noperator" dx(t)/dt - which should be an "observable". Indeed, we can\nmeasure velocities of particles. But when you compute it, you see that\nis proportional to id/dx - A. But A is an electromagnetic potential\nthat, in itself is not an observable, because it can be replaced by A+d\nlambda with the same physical implications. Therefore i d/dx must not be\nan observable either! The only solution out of the dilemma is:\n\nwhenever you replace A by A+ d lambda, replace your wave function psi by\nexp(i lambda) psi, so that the two contributions from d/dx and A will\ncancel and velocity will be unchanged.\n\nark\n--\n\nArkadiusz Jadczyk\nhttp://quantumfuture.net/quantum_future/jadpub.htm\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>On 14 Aug 2004 07:58:04 -0400, "Flip Tomato" <flipt@stanford.edu> wrote:

>
>What motivates the definition of the gauge covariant derivative, other than
>that it gives a nice result?

There are many possible answers to this question. Let me propose one
that appeals to me.

Consider quantum particle in the external EM field. Compute "velocity
operator" dx(t)/dt - which should be an "observable". Indeed, we can
measure velocities of particles. But when you compute it, you see that
is proportional to id/dx - A. But A is an electromagnetic potential
that, in itself is not an observable, because it can be replaced by A+d
\lambda with the same physical implications. Therefore i d/dx must not be
an observable either! The only solution out of the dilemma is:

whenever you replace A by A+ d \lambda, replace your wave function \psi by
\exp(i \lambda) \psi, so that the two contributions from d/dx and A will
cancel and velocity will be unchanged.

ark
--

Arkadiusz Jadczyk
http://quantumfuture.net/quantum_future/jadpub.htm
--

Igor

Aug16-04, 12:56 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>\n\n\n\n"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cfjhcp\\$scv\\$1@news.Stanford.EDU>...\n> Greetings--I\'m trying to do some reading into relativistic quantum mechanics\n> (I\'ve just taken quantum at the undergrad level) and I\'m curious about the\n> covariant derivative that is used when discussing gauge invariance.\n>\n> What motivates the definition of the gauge covariant derivative, other than\n> that it gives a nice result? I understand the gradient (1) mathematically as\n> a derivative operator and (2) physically as a momentum operator--but the\n> gauge covariant derivative doesn\'t have any intuitive appeal to me other\n> than sketchily looking like it would involve the gauge freedom of choosing\n> A.\n>\n> In Chris Quigg\'s "Gauge Theories of the..." book, he writes: "Local phase\n> invariance may be achieved if the equations of motion and the observables\n> involving derivatives are modified by the introduction of the\n> electromagnetic field A_\\mu(x). If the gradient is everywhere replaced by\n> the gauge covariant derivative, [this is satisfied]."\n>\n> In terms of the big picture, I understand (but please correct me if i\'m\n> wrong) that local phase invariance is a general principle that we would like\n> to have in quantum mechanics, so we impose it by introducing this gauge\n> covariant derivative. The term in the gauge covariatn derivative, A_\\mu(x),\n> then *turns out* to be the EM potential and lo and behold, E&M pops out of\n> this principle of local phase invariance. The fact that E&M naturally pops\n> out of this principle--this is "evidence" to believe that local phase\n> invariance is a reasonable "first principle"?\n>\n> I know my questions are a little hazy right now as I\'m still trying to get\n> my head around these topics--but any insight would be much appreciated (and\n> probably followed by more precise questions).\n>\n> Thanks,\n> Flip Tanedo\n> flipt (at) stanford (dot) edu\n>\n> PS--on a totally unrelated note, I\'m not very good with literature searches\n> yet... how do I find *review* articles in a subject that I\'m interested in\n> studying?\n\n\nThe whole point of a covariant derivative, be it in EM gauge theory or\nin differential geometry as applied to GR, is to literally keep the\ndifferential operator covariant. In other words, making sure that the\ngradient remains a true vector operator under transformations. A true\nvector must transform in the form v\' = M v, where v\' and v are column\nvectors, and M is a square matrix. Quite often, the ordinary\nderivative operator will not transform in this way and requires\nadditional terms to bring it back in line with the notion of\ncovariance. That\'s really all there is to it.\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>"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cfjhcp$scv$1@news.Stanford.EDU>...
> Greetings--I'm trying to do some reading into relativistic quantum mechanics
> (I've just taken quantum at the undergrad level) and I'm curious about the
> covariant derivative that is used when discussing gauge invariance.
>
> What motivates the definition of the gauge covariant derivative, other than
> that it gives a nice result? I understand the gradient (1) mathematically as
> a derivative operator and (2) physically as a momentum operator--but the
> gauge covariant derivative doesn't have any intuitive appeal to me other
> than sketchily looking like it would involve the gauge freedom of choosing
> A.
>
> In Chris Quigg's "Gauge Theories of the..." book, he writes: "Local phase
> invariance may be achieved if the equations of motion and the observables
> involving derivatives are modified by the introduction of the
> electromagnetic field A_\mu(x). If the gradient is everywhere replaced by
> the gauge covariant derivative, [this is satisfied]."
>
> In terms of the big picture, I understand (but please correct me if i'm
> wrong) that local phase invariance is a general principle that we would like
> to have in quantum mechanics, so we impose it by introducing this gauge
> covariant derivative. The term in the gauge covariatn derivative, A_\mu(x),
> then *turns out* to be the EM potential and lo and behold, E&M pops out of
> this principle of local phase invariance. The fact that E&M naturally pops
> out of this principle--this is "evidence" to believe that local phase
> invariance is a reasonable "first principle"?
>
> I know my questions are a little hazy right now as I'm still trying to get
> my head around these topics--but any insight would be much appreciated (and
> probably followed by more precise questions).
>
> Thanks,
> Flip Tanedo
> flipt (at) stanford (dot) edu
>
> PS--on a totally unrelated note, I'm not very good with literature searches
> yet... how do I find *review* articles in a subject that I'm interested in
> studying?

The whole point of a covariant derivative, be it in EM gauge theory or
in differential geometry as applied to GR, is to literally keep the
differential operator covariant. In other words, making sure that the
gradient remains a true vector operator under transformations. A true
vector must transform in the form v' = M v, where v' and v are column
vectors, and M is a square matrix. Quite often, the ordinary
derivative operator will not transform in this way and requires
additional terms to bring it back in line with the notion of
covariance. That's really all there is to it.

mandro

Aug16-04, 12:56 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>\n\n\n\n"Flip Tomato" ,\n\nAs I have understood this, I think that the story is\nsomething like this: A connection "A" defines a "parallel\ntransport"(In some sense a parallel transport can be seen\nas a prescription that given a parametrized curve and a\nvector w on one end of the curve, assigns to this a unique\nvector field along the curve (where of course, the vector\nthat the vector field assigns to the endpoint is w.). Now,\nonce we have this, we can parallel transport n basis basis\nvectors along any curve , and express any vector field along\nthe curve in terms of this basis i.e., with components C_i,\nthen take the derivative of the components C_i(t) and that\nvector field is the covariant derivative. Ugh... I think\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>"Flip Tomato" ,

As I have understood this, I think that the story is
something like this: A connection "A" defines a "parallel
transport"(In some sense a parallel transport can be seen
as a prescription that given a parametrized curve and a
vector w on one end of the curve, assigns to this a unique
vector field along the curve (where of course, the vector
that the vector field assigns to the endpoint is w.). Now,
once we have this, we can parallel transport n basis basis
vectors along any curve , and express any vector field along
the curve in terms of this basis i.e., with components C_i,
then take the derivative of the components C_i(t) and that
vector field is the covariant derivative. Ugh... I think

Danny Ross Lunsford

Aug17-04, 11:27 AM

<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>\n\n\n\n"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cfjhcp\\$scv\\$1@news.Stanford.EDU>...\n> Greetings--I\'m trying to do some reading into relativistic quantum mechanics\n> (I\'ve just taken quantum at the undergrad level) and I\'m curious about the\n> covariant derivative that is used when discussing gauge invariance.\n>\n> What motivates the definition of the gauge covariant derivative, other than\n> that it gives a nice result?\n\nWeyl developed the idea while thinking about gravity. The latter has a\nglobal symmetry, length scale. Weyl developed a geometry in which both\nlength and direction were strictly local - he called this "pure\ninfinitesimal geometry". The localization of the formerly global\nlength symmetry shows up as a new field alongside the metric, which\nWeyl interpreted as the electromagnetic potential. Alongside the\ngeneral coordinate transformations at each point in spacetime one also\nnow has literal gauge transformations, that is, recalibration of the\nlocal length scale. Under such a transformation, the Weyl gauge field\nAm and the metric gmn change as\n\ngmn -> exp(L) gmn\n\nAm -> Am - d/dxm L\n\nUnder coordinate transformations the Am are just a covariant vector.\nThus exactly the right number of new fields are introduced, along with\nthe right transformation properties, to make a joint theory of light\nand gravity. The conservation of energy-momentum and electric charge\nare on the same logical footing in this theory.\n\nIn Weyl\'s geometry the covariant derivative of a tensor takes the form\n\nDm Tab.. = dm Tab.. + N Am Tab..\n\nwhere N is the "conformal weight" of the tensor Tab.. and dm is the\nusual Riemannian covariant derivative.\n\nThe theory failed for both physical and mathematical reasons - there\nis no sensible Lagrangian theory leading to second order equations for\nthe gmn that are irreducibly coupled to the Am. Spacetime is just the\nwrong dimension to make the idea workable (the spacetime dimension is\nimportant because the volume element transforms differently depending\non the dimension). However, when the Dirac electron theory was\nproposed, Weyl immediately saw that the arbitrary phase of the\nelectron field is of exactly the same nature as the arbitrary length\nscale in his geometry, and is also associated with the conservation of\ncharge in nearly the same way as before - only now the covariant\nderivative is applied to spinors with the form\n\nDm = dm + ie Am\n\nso in a sense, in Dirac\'s theory the spinor field has "imaginary\nweight" ie.\n\nTo sum up, the term "gauge" literally meant localized length, and\n"covariant derivative" was a concept taken directly from Weyl\'s\nmodification of Ricci\'s "absolute differential calculus".\n\nNowadays of course the mathematicians have created the theory of fiber\nbundles to systematize these connections and their covariant\nderivatives.\n\n-drl\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>"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cfjhcp$scv$1@news.Stanford.EDU>...
> Greetings--I'm trying to do some reading into relativistic quantum mechanics
> (I've just taken quantum at the undergrad level) and I'm curious about the
> covariant derivative that is used when discussing gauge invariance.
>
> What motivates the definition of the gauge covariant derivative, other than
> that it gives a nice result?

Weyl developed the idea while thinking about gravity. The latter has a
global symmetry, length scale. Weyl developed a geometry in which both
length and direction were strictly local - he called this "pure
infinitesimal geometry". The localization of the formerly global
length symmetry shows up as a new field alongside the metric, which
Weyl interpreted as the electromagnetic potential. Alongside the
general coordinate transformations at each point in spacetime one also
now has literal gauge transformations, that is, recalibration of the
local length scale. Under such a transformation, the Weyl gauge field
Am and the metric gmn change as

gmn -> \exp(L) gmn

Am -> Am - d/dxm L

Under coordinate transformations the Am are just a covariant vector.
Thus exactly the right number of new fields are introduced, along with
the right transformation properties, to make a joint theory of light
and gravity. The conservation of energy-momentum and electric charge
are on the same logical footing in this theory.

In Weyl's geometry the covariant derivative of a tensor takes the form

Dm Tab.. = dm Tab.. + N Am Tab..

where N is the "conformal weight" of the tensor Tab.. and dm is the
usual Riemannian covariant derivative.

The theory failed for both physical and mathematical reasons - there
is no sensible Lagrangian theory leading to second order equations for
the gmn that are irreducibly coupled to the Am. Spacetime is just the
wrong dimension to make the idea workable (the spacetime dimension is
important because the volume element transforms differently depending
on the dimension). However, when the Dirac electron theory was
proposed, Weyl immediately saw that the arbitrary phase of the
electron field is of exactly the same nature as the arbitrary length
scale in his geometry, and is also associated with the conservation of
charge in nearly the same way as before - only now the covariant
derivative is applied to spinors with the form

Dm = dm + ie Am

so in a sense, in Dirac's theory the spinor field has "imaginary
weight" ie.

To sum up, the term "gauge" literally meant localized length, and
"covariant derivative" was a concept taken directly from Weyl's
modification of Ricci's "absolute differential calculus".

Nowadays of course the mathematicians have created the theory of fiber
bundles to systematize these connections and their covariant
derivatives.

-drl