View Full Version : why graviton's sipn = 2
Alexander Gorshenev
Apr24-04, 11:16 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>I have read multiple times that graviton should have\nspin = 2.\n\nIs this just the conclusion based on never\nobserving the gravitational repulsion or there are some\nmore deep reasons for such statement?\n\nA.G.\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>I have read multiple times that graviton should have
spin = 2.
Is this just the conclusion based on never
observing the gravitational repulsion or there are some
more deep reasons for such statement?
A.G.
Danny Ross Lunsford
Apr24-04, 08:16 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>Alexander Gorshenev wrote:\n\n> I have read multiple times that graviton should have\n> spin = 2.\n>\n> Is this just the conclusion based on never\n> observing the gravitational repulsion or there are some\n> more deep reasons for such statement?\n\nIt is based on the kinematics of representations of the Lorentz group,\ncombined with an analysis of gravitational radiation. This radiation is\ndescribed by a polarization tensor that has, according to the general\ntheory, spin-2, spin-1, and spin-0 components. The latter two can be\nmade to vanish by a coordinate transformation. Thus the "essential" part\nof the radiation is the spin-2 part. The key idea is the masslessness of\nthe radiation, which allows it to be invariantly characterized by its\nhelicity. For massless particles the spin is parallel or anti-parallel\nto the momentum, and this is invariant under a Lorentz transformation.\n\nA similar thing happens with electromagnetic radiation, which, according\nto the general theory, is composed of spin-1 and spin-0 parts. The\nlatter can be made to vanish by a gauge transformation. The "essential"\npart of the radiation is spin-1.\n\nNote that no quantization is implied here - this is simply the\nkinematics of representation theory.\n\n-drl\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>Alexander Gorshenev wrote:
> I have read multiple times that graviton should have
> spin = 2.
>
> Is this just the conclusion based on never
> observing the gravitational repulsion or there are some
> more deep reasons for such statement?
It is based on the kinematics of representations of the Lorentz group,
combined with an analysis of gravitational radiation. This radiation is
described by a polarization tensor that has, according to the general
theory, spin-2, spin-1, and spin-0 components. The latter two can be
made to vanish by a coordinate transformation. Thus the "essential" part
of the radiation is the spin-2 part. The key idea is the masslessness of
the radiation, which allows it to be invariantly characterized by its
helicity. For massless particles the spin is parallel or anti-parallel
to the momentum, and this is invariant under a Lorentz transformation.
A similar thing happens with electromagnetic radiation, which, according
to the general theory, is composed of spin-1 and spin-0 parts. The
latter can be made to vanish by a gauge transformation. The "essential"
part of the radiation is spin-1.
Note that no quantization is implied here - this is simply the
kinematics of representation theory.
-drl
Danny Ross Lunsford
Apr27-04, 01:47 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>Alexander Gorshenev wrote:\n\n> I have read multiple times that graviton should have\n> spin = 2.\n>\n> Is this just the conclusion based on never\n> observing the gravitational repulsion or there are some\n> more deep reasons for such statement?\n\nIt is based on the kinematics of representations of the Lorentz group,\ncombined with an analysis of gravitational radiation. This radiation is\ndescribed by a polarization tensor that has, according to the general\ntheory, spin-2, spin-1, and spin-0 components. The latter two can be\nmade to vanish by a coordinate transformation. Thus the "essential" part\nof the radiation is the spin-2 part. The key idea is the masslessness of\nthe radiation, which allows it to be invariantly characterized by its\nhelicity. For massless particles the spin is parallel or anti-parallel\nto the momentum, and this is invariant under a Lorentz transformation.\n\nA similar thing happens with electromagnetic radiation, which, according\nto the general theory, is composed of spin-1 and spin-0 parts. The\nlatter can be made to vanish by a gauge transformation. The "essential"\npart of the radiation is spin-1.\n\nNote that no quantization is implied here, other than that of angular\nmomentum - this is simply the kinematics of representation theory. That\nis, this analysis is correct regardless of the details of quantum gravity.\n\n-drl\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>Alexander Gorshenev wrote:
> I have read multiple times that graviton should have
> spin = 2.
>
> Is this just the conclusion based on never
> observing the gravitational repulsion or there are some
> more deep reasons for such statement?
It is based on the kinematics of representations of the Lorentz group,
combined with an analysis of gravitational radiation. This radiation is
described by a polarization tensor that has, according to the general
theory, spin-2, spin-1, and spin-0 components. The latter two can be
made to vanish by a coordinate transformation. Thus the "essential" part
of the radiation is the spin-2 part. The key idea is the masslessness of
the radiation, which allows it to be invariantly characterized by its
helicity. For massless particles the spin is parallel or anti-parallel
to the momentum, and this is invariant under a Lorentz transformation.
A similar thing happens with electromagnetic radiation, which, according
to the general theory, is composed of spin-1 and spin-0 parts. The
latter can be made to vanish by a gauge transformation. The "essential"
part of the radiation is spin-1.
Note that no quantization is implied here, other than that of angular
momentum - this is simply the kinematics of representation theory. That
is, this analysis is correct regardless of the details of quantum gravity.
-drl
<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>horsh@mail.ru (Alexander Gorshenev) wrote in message news:<abb5d97a.0404220918.14689fdc@posting.google. com>...\n> I have read multiple times that graviton should have\n> spin = 2.\n>\n> Is this just the conclusion based on never\n> observing the gravitational repulsion or there are some\n> more deep reasons for such statement?\n>\n> A.G.\n\nThe simplest way to approach this question involves the properties of\nthe particular field under rotations. The field f transforms as f\' =\nf exp (i n @), where @ is the rotation angle and n is the spin quantum\nnumber of the field. The actual spin angular momentum is given by S =\nn hbar. Scalar particles, such as the hypothetical Higgs, have n = 0,\nsince scalar fields are invariant under rotations. Hence scalar\nparticles have no spin. Vector particles, such as photons or weak\nbosons, have n = 1, since this corresponds to the transformation\nproperties of a vector under rotations. The best way to picture this\nis a vector in two dimensional space whose components correspond to\nthe real and imaginary parts of a single complex number.\n\nFor higher order objects, we can think of them as being composed of a\nspecial product of vectors (usually called the direct or tensor\nproduct). Such a product of two vectors is usually referred to as\nsecond rank tensor, although not all tensors are reducible in this\nway. This is the basis of gravitation (all other inertial\naccelerations) in General Relativity. Since each vector field degree\nof freedom provides spin n = 1, the two of them taken together have\nspin n = 2. Theoretically this analysis could continue forever ad\ninfinitum. There could, in theory, be particle tensor fields of mth\nrank corresponding to spin n = m.\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>horsh@mail.ru (Alexander Gorshenev) wrote in message news:<abb5d97a.0404220918.14689fdc@posting.google.com>...
> I have read multiple times that graviton should have
> spin = 2.
>
> Is this just the conclusion based on never
> observing the gravitational repulsion or there are some
> more deep reasons for such statement?
>
> A.G.
The simplest way to approach this question involves the properties of
the particular field under rotations. The field f transforms as f' =f \exp (i n @), where @ is the rotation angle and n is the spin quantum
number of the field. The actual spin angular momentum is given by S =
n \hbar. Scalar particles, such as the hypothetical Higgs, have n = 0,
since scalar fields are invariant under rotations. Hence scalar
particles have no spin. Vector particles, such as photons or weak
bosons, have n = 1, since this corresponds to the transformation
properties of a vector under rotations. The best way to picture this
is a vector in two dimensional space whose components correspond to
the real and imaginary parts of a single complex number.
For higher order objects, we can think of them as being composed of a
special product of vectors (usually called the direct or tensor
product). Such a product of two vectors is usually referred to as
second rank tensor, although not all tensors are reducible in this
way. This is the basis of gravitation (all other inertial
accelerations) in General Relativity. Since each vector field degree
of freedom provides spin n = 1, the two of them taken together have
spin n = 2. Theoretically this analysis could continue forever ad
infinitum. There could, in theory, be particle tensor fields of mth
rank corresponding to spin n = m.
Arnold Neumaier
Apr28-04, 01:46 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>Alexander Gorshenev wrote:\n> I have read multiple times that graviton should have\n> spin = 2.\n>\n> Is this just the conclusion based on never\n> observing the gravitational repulsion or there are some\n> more deep reasons for such statement?\n\nThe reason is that gravitation is described by a metric\n(symmetric 2-tensor field) modulo general covariance,\nwhich gives [locally in the tangent Minkowski space of any point]\na spin 2 representation of the Poincare group.\nGravitational waves have to be (classically) long range,\nwhich requires (after quantization) massless particles.\nThus gravitons (although never observed) should be massless\nspin 2 particles.\n\n\nArnold Neumaier\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>Alexander Gorshenev wrote:
> I have read multiple times that graviton should have
> spin = 2.
>
> Is this just the conclusion based on never
> observing the gravitational repulsion or there are some
> more deep reasons for such statement?
The reason is that gravitation is described by a metric
(symmetric 2-tensor field) modulo general covariance,
which gives [locally in the tangent Minkowski space of any point]
a spin 2 representation of the Poincare group.
Gravitational waves have to be (classically) long range,
which requires (after quantization) massless particles.
Thus gravitons (although never observed) should be massless
spin 2 particles.
Arnold Neumaier
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