[SOLVED] on shell, off shell - what does it mean [Archive]

Ted Sung

Jun27-04, 05:58 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>Hi,\n\nI see the terms "on shell" and "off shell" used from time to time.\nWhat do they mean and how are they used?\n\nThanks,\n\nTed\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>Hi,

I see the terms "on shell" and "off shell" used from time to time.
What do they mean and how are they used?

Thanks,

Ted

Aaron Bergman

Jun29-04, 05:41 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>In article <86dea8f5.0406210532.56b82380@posting.google.com>, \nteds@intex.com (Ted Sung) wrote:\n\n> Hi,\n>\n> I see the terms "on shell" and "off shell" used from time to time.\n> What do they mean and how are they used?\n\n"on shell" = obeying the equations of motion\n"off shell" = not obeying the equations of motion\n\nAaron\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>In article <86dea8f5.0406210532.56b82380@posting.google.com>,
teds@intex.com (Ted Sung) wrote:

> Hi,
>
> I see the terms "on shell" and "off shell" used from time to time.
> What do they mean and how are they used?

"on shell" = obeying the equations of motion
"off shell" = not obeying the equations of motion

Aaron

Hendrik van Hees

Jun29-04, 05:42 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>Ted Sung wrote:\n\n> Hi,\n>\n> I see the terms "on shell" and "off shell" used from time to time.\n> What do they mean and how are they used?\n\nIn (vacuum) quantum field theory, the physical observables, we like to\ncalculate are S-matrix elements, from which we can make contact to the\nexperimenters\' results like life times of particles or cross sections\nfor certain events.\n\nSince we are so far too stupid to do otherwise, we use perturbation\ntheory to evaluate these quantities from quantum field theory, and we\nhave a nice tool to handle these calculations: Feynman diagrams.\n\nFirst, each Feynman diagram stands for an S-matrix amplitude. There we\nconsider particles, we prepare in certain states such that we can\nconsider them to be free particles. Then we let them run against each\nother, so that they become interacting and put a detector somewhere far\naway from the reaction zone, such that again we observe particles,\nwhich can be considered free. In short words, we observe asymptotically\nfree particles (that\'s also slang, you\'ll read often in qft text\nbooks).\n\nThus the external legs of Feynman diagrams stand for asymptotically free\nstates, and each particle is then taken to be on its mass shell or, in\nshort slang, "on shell". The word comes from the shape of the\nenergy-momentum relation\n\nE^2 -p^2/c^2=m^2 c^2\n\nof the free particles which is a hyperboloid in momentum space (m=const\nof course).\n\nNow, the experimenters measure with higher and higher accuracy,\nchallenging the theoreticians to go to higher orders in perturbation\ntheory. Then another great feature of the Feynman diagrams helps a lot!\nThey can be decomposed in simpler parts. You\'ll also find this in qft\ntext books, there are all kinds of diagram classes around, among which\nthe most important are the amputated one-particle irreducible Green\'s\nfunctions, also known as proper vertex functions. They might be\ncomplicated to calculate, but whenever you have them you can use them\nas building blocks for higher-order Feynman diagrams, which represent\nscattering amplitudes.\n\nSince most of the time these functions have to be inserted in the\ninterior of the Feynman diagrams, we like to calculate, their external\nlegs (which are amputated anyway ;-)) are not to be taken on shell.\nThat\'s consequently called "off the mass shell" or briefly "off shell".\n\nThe interior lines of Feynman diagrams represent the fields, causing the\ninteraction. Often they are also entitled virtual particles which are\n"exchanged" by the real particles and that makes them interact, but one\nshouldn\'t take this too literary, just as a picture to memorise Feynman\ndiagrams.\n\nOn the other hand, the "virtual particles", which you should better\nthink of fields, which according to quantum theory can fluctuate. Even\nin the vacuum there are vacuum fluctuations of the fields present, and\nthe vacuum is not just void in the classical sense of the word. The\neffect of vacuum fluctuations can be measured. This phenomenon is known\nas the Casimir effect.\n\nFor instance, if one considers two uncharged metal plates, these plates\nset boundary conditions for the electromagnetic field. In classical\nMaxwell theory, there is no field present, because the plates are not\ncharged. Nevertheless due to Heisenberg\'s uncertainty relation, the\nelectromagnetic field is fluctuating: Although it\'s expectation value\nvanishes in accordance with classical electromagnetism, its\nuncertainties (or standard deviations in the sense of statistics)\ndon\'t. Since now the fluctuating fields have less states inside the\nplates than outside, these quantum fluctuations of the electromagnetic\nfields make a net effect of a very weak attractive force between the\nplates, the Casimir force, which is measured nowadays.\n\nYou find an elementary calculation of this effect in\n\nItzykson, Zuber, Quantum Field Theory, McGraw Hill\n\nA reference to a recent experiment is\n\nS. K. Lamoreaux, Demonstration of the Casimir Force in the 0.6 to 6 mum\nRange, Phys. Rev. Lett. *78* (1997) _5_\n\nSee also the comment in PRL *84* (2000) _5672_ and references therein.\n\n--\nHendrik van Hees Cyclotron Institute\nPhone: +1 979/845-1411 Texas A&M University\nFax: +1 979/845-1899 Cyclotron Institute, MS-3366\nhttp://theory.gsi.de/~vanhees/ College Station, TX 77843-3366\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>Ted Sung wrote:

> Hi,
>
> I see the terms "on shell" and "off shell" used from time to time.
> What do they mean and how are they used?

In (vacuum) quantum field theory, the physical observables, we like to
calculate are S-matrix elements, from which we can make contact to the
experimenters' results like life times of particles or cross sections
for certain events.

Since we are so far too stupid to do otherwise, we use perturbation
theory to evaluate these quantities from quantum field theory, and we
have a nice tool to handle these calculations: Feynman diagrams.

First, each Feynman diagram stands for an S-matrix amplitude. There we
consider particles, we prepare in certain states such that we can
consider them to be free particles. Then we let them run against each
other, so that they become interacting and put a detector somewhere far
away from the reaction zone, such that again we observe particles,
which can be considered free. In short words, we observe asymptotically
free particles (that's also slang, you'll read often in qft text
books).

Thus the external legs of Feynman diagrams stand for asymptotically free
states, and each particle is then taken to be on its mass shell or, in
short slang, "on shell". The word comes from the shape of the
energy-momentum relation

E^2 -p^2/c^2=m^2 c^2

of the free particles which is a hyperboloid in momentum space (m=const
of course).

Now, the experimenters measure with higher and higher accuracy,
challenging the theoreticians to go to higher orders in perturbation
theory. Then another great feature of the Feynman diagrams helps a lot!
They can be decomposed in simpler parts. You'll also find this in qft
text books, there are all kinds of diagram classes around, among which
the most important are the amputated one-particle irreducible Green's
functions, also known as proper vertex functions. They might be
complicated to calculate, but whenever you have them you can use them
as building blocks for higher-order Feynman diagrams, which represent
scattering amplitudes.

Since most of the time these functions have to be inserted in the
interior of the Feynman diagrams, we like to calculate, their external
legs (which are amputated anyway ;-)) are not to be taken on shell.
That's consequently called "off the mass shell" or briefly "off shell".

The interior lines of Feynman diagrams represent the fields, causing the
interaction. Often they are also entitled virtual particles which are
"exchanged" by the real particles and that makes them interact, but one
shouldn't take this too literary, just as a picture to memorise Feynman
diagrams.

On the other hand, the "virtual particles", which you should better
think of fields, which according to quantum theory can fluctuate. Even
in the vacuum there are vacuum fluctuations of the fields present, and
the vacuum is not just void in the classical sense of the word. The
effect of vacuum fluctuations can be measured. This phenomenon is known
as the Casimir effect.

For instance, if one considers two uncharged metal plates, these plates
set boundary conditions for the electromagnetic field. In classical
Maxwell theory, there is no field present, because the plates are not
charged. Nevertheless due to Heisenberg's uncertainty relation, the
electromagnetic field is fluctuating: Although it's expectation value
vanishes in accordance with classical electromagnetism, its
uncertainties (or standard deviations in the sense of statistics)
don't. Since now the fluctuating fields have less states inside the
plates than outside, these quantum fluctuations of the electromagnetic
fields make a net effect of a very weak attractive force between the
plates, the Casimir force, which is measured nowadays.

You find an elementary calculation of this effect in

Itzykson, Zuber, Quantum Field Theory, McGraw Hill

A reference to a recent experiment is

S. K. Lamoreaux, Demonstration of the Casimir Force in the .6 to 6 mum
Range, Phys. Rev. Lett. *78* (1997) _5_

See also the comment in PRL *84* (2000) _5672_ and references therein.

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

Igor Khavkine

Jun29-04, 05:43 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>teds@intex.com (Ted Sung) wrote in message news:<86dea8f5.0406210532.56b82380@posting.google. com>...\n> Hi,\n>\n> I see the terms "on shell" and "off shell" used from time to time.\n> What do they mean and how are they used?\n\nThis terminology usually comes up when discussing scattering amplitudes\nin particle physics (which are calculated using Feynman diagrams). These\namplitudes depend on the momenta of incoming and outgoing particles\n(or the momentum labels on the legs of the Feynman diagrams).\n\nIn reality, the 4-momenta of the incoming and outgoing particles are\nnot arbitrary, but are constrained by the relation E^2 - p^2 = m^2,\nwhere m is the mass of the particle, E is its total energy, and p the\nspacial momentum. A 4-momentum that satisfies this constraint is called\n"on shell", for the reason that it must lie on a "shell" in 4-momentum\nspace defined by the constraint. However, for some theoretical\ninvestigations, it is convenient or necessary to consider arbitrary\n4-momenta that are not constrained by the above relation and hence are\ncalled "off shell".\n\nHope this helps.\n\nIgor\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>teds@intex.com (Ted Sung) wrote in message news:<86dea8f5.0406210532.56b82380@posting.google.com>...
> Hi,
>
> I see the terms "on shell" and "off shell" used from time to time.
> What do they mean and how are they used?

This terminology usually comes up when discussing scattering amplitudes
in particle physics (which are calculated using Feynman diagrams). These
amplitudes depend on the momenta of incoming and outgoing particles
(or the momentum labels on the legs of the Feynman diagrams).

In reality, the 4-momenta of the incoming and outgoing particles are
not arbitrary, but are constrained by the relation E^2 - p^2 = m^2,
where m is the mass of the particle, E is its total energy, and p the
spacial momentum. A 4-momentum that satisfies this constraint is called
"on shell", for the reason that it must lie on a "shell" in 4-momentum
space defined by the constraint. However, for some theoretical
investigations, it is convenient or necessary to consider arbitrary
4-momenta that are not constrained by the above relation and hence are
called "off shell".

Hope this helps.

Igor

Thomas Dent

Jun29-04, 05:55 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>teds@intex.com (Ted Sung) wrote\n\n> Hi,\n>\n> I see the terms "on shell" and "off shell" used from time to time.\n> What do they mean and how are they used?\n>\n> Thanks,\n>\n> Ted\n\n\nShort for \'on mass shell\' and \'off mass shell\'\n\nThe \'mass shell\' for a particle of mass m is the surface in the E - p1\n- p2 - p3 space satisfying the equation\n\nE^2 - p1^2 - p2^2 - p3^2 = m^2 (factors of c omitted)\n\nIt\'s kind of a hyperboloid in 4 dimensional space.\n\nOr if P is the energy-momentum 4-vector (E, p1, p2, p3),\n\nP.P = m^2\n\n(You may see this with the opposite sign of P.P depending on the\nmetric.)\n\nIn classical special relativity all particles are \'on mass shell\'. For\np=0 we recover the infamous\n\nE^2 = m^2\n\nor with factors of c replaced\n\nE^2 = m^2 c^4\n\nIn quantum field theory particles are allowed to propagate even if\nthey do not satisfy this equation. In that case they are said to be\n\'off mass shell\'. These are also called \'virtual particles\'. However,\nthere is a penalty to pay for this, because in that case the amplitude\nof interaction goes as\n\n1/(P.P - m^2)\n\nParticles that can propagate to infinity (\'real particles\') are always\n\'on mass shell\'. In a collider experiment, a few centimetres is a\nreasonable approximation to infinity, so one takes the incoming and\noutgoing particles \'on shell\'. Intermediate states are \'off shell\' or\n\'virtual\'.\n\nAny further questions?\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>teds@intex.com (Ted Sung) wrote

> Hi,
>
> I see the terms "on shell" and "off shell" used from time to time.
> What do they mean and how are they used?
>
> Thanks,
>
> Ted

Short for 'on mass shell' and 'off mass shell'

The 'mass shell' for a particle of mass m is the surface in the E - p1- p2 - p3 space satisfying the equation

E^2 - p1^2 - p2^2 - p3^2 = m^2[/itex] (factors of c omitted)

It's kind of a hyperboloid in 4 dimensional space.

Or if P is the energy-momentum 4-vector [itex](E, p1, p2, p3),P.P = m^2

(You may see this with the opposite sign of P.P depending on the
metric.)

In classical special relativity all particles are 'on mass shell'. For
p=0 we recover the infamous

E^2 = m^2

or with factors of c replaced

E^2 = m^2 c^4

In quantum field theory particles are allowed to propagate even if
they do not satisfy this equation. In that case they are said to be
'off mass shell'. These are also called 'virtual particles'. However,
there is a penalty to pay for this, because in that case the amplitude
of interaction goes as

1/(P.P - m^2)

Particles that can propagate to infinity ('real particles') are always
'on mass shell'. In a collider experiment, a few centimetres is a
reasonable approximation to infinity, so one takes the incoming and
outgoing particles 'on shell'. Intermediate states are 'off shell' or
'virtual'.

Any further questions?

arivero

Jun29-04, 05:56 PM

Ted, an off-shell particle is in violation of the relativistic energy equation E^2=m^2c^4+p^2c^2.

Usually such off-shell objects are considered only when adding all the possible paths for a transition.
Also, a naive view is that E can be violated during a short time interval h/(Delta E), or similarly p can be violated along a small volume.

Aleksey Kovalenko

Jun30-04, 05:38 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>teds@intex.com (Ted Sung) wrote in message news:<86dea8f5.0406210532.56b82380@posting.google. com>...\n> Hi,\n>\n> I see the terms "on shell" and "off shell" used from time to time.\n> What do they mean and how are they used?\n>\n> Thanks,\n>\n> Ted\n\nHi\nIn QFT "On [mass] shell" particle obeys classical equations of motion\n(e.g. p^2=m^2 for scalar and so on).\nAleksey\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>teds@intex.com (Ted Sung) wrote in message news:<86dea8f5.0406210532.56b82380@posting.google.com>...
> Hi,
>
> I see the terms "on shell" and "off shell" used from time to time.
> What do they mean and how are they used?
>
> Thanks,
>
> Ted

Hi
In QFT "On [mass] shell" particle obeys classical equations of motion
(e.g. p^2=m^2 for scalar and so on).
Aleksey

Arnold Neumaier

Jun30-04, 10:04 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>Ted Sung wrote:\n\n> I see the terms "on shell" and "off shell" used from time to time.\n> What do they mean and how are they used?\n\nI added the following to my theoretical physics FAQ at\nhttp://www.mat.univie.ac.at/~neum/physics-faq.txt\n\n\nArnold Neumaier\n\n\n==================================== ========================================\n\n\n\n--------------------------------------------------\nWhat is the meaning of \'on-shell\' and \'off-shell\'?\n--------------------------------------------------\n\nThis applies only to relativistic particles.\nA particle of mass m is on-shell if its momentum p satisfies\np^2 (= p_0^2-p_1^2-p_2^2-p_3^2) = m^2,\nand off-shell otherwise. The \'mass shell\' is the manifold of\nmomenta p with p^2=m^2.\n\nObservable (i.e., physical) particles are asymptotic states\n(scattering states) described (modulo unresolved mathematical\ndifficulties) by free fields based on the dispersion relation p^2=m^2,\nand hence are necessarily on-shell. Off-shell particles only\narise in intermediate perturbative calculations; they are necessarily\n\'virtual\'.\n\nThe situation is muddled by the fact that one has to distinguish\n(formal) bare mass and (physical) dressed mass; the above is valid\nonly for the dressed mass. Moreover, the mass shell loses its meaning\nin external fields, where, instead, a so-called \'gap equation\'\nappears.\n\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>Ted Sung wrote:

> I see the terms "on shell" and "off shell" used from time to time.
> What do they mean and how are they used?

I added the following to my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt

Arnold Neumaier

================================================== ==========================

--------------------------------------------------
What is the meaning of 'on-shell' and 'off-shell'?
--------------------------------------------------

This applies only to relativistic particles.
A particle of mass m is on-shell if its momentum p satisfies
p^2 (= p_0^2-p_1^2-p_2^2-p_3^2) = m^2,
and off-shell otherwise. The 'mass shell' is the manifold of
momenta p with p^2=m^2.

Observable (i.e., physical) particles are asymptotic states
(scattering states) described (modulo unresolved mathematical
difficulties) by free fields based on the dispersion relation p^2=m^2,
and hence are necessarily on-shell. Off-shell particles only
arise in intermediate perturbative calculations; they are necessarily
'virtual'.

The situation is muddled by the fact that one has to distinguish
(formal) bare mass and (physical) dressed mass; the above is valid
only for the dressed mass. Moreover, the mass shell loses its meaning
in external fields, where, instead, a so-called 'gap equation'
appears.