## Yang-Mills Fields

<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>Hello,\n\nGiven a complete manifold X, Let d be the de Rham operator and A a 1\nform (with value in R). The magnetic laplacian is defined as\n\\Delta_A=(d+iA)*(d+iA).\n\nIn the general case, let be E be vector bundle on X. Let \\nabla be a\nconnection on E. I define the "Yang-Mills Laplacian" as being\n\\Delta_YM=\\nabla*\\nabla.\n\nLet E be associated to a principal bundle with group G, i.e. E is\nlocally XxF where G acts on F.\n\n1. Is the "Yang-Mills Laplacian" of physical interest? (I guess so\nbecause it generalizes the magnetic laplacian)\n\n2. Are there special G and representation F of particular interest?\n\n3. If the cohomology group H^1 of X is trivial, we have gauge\ninvariance for the magnetic laplacian, i.e. it depends, up to unitary\nequivalence, only on the curvature dA of the connection. What is the\nanalogous property for the YM Laplacian ?\n\nThanks for your help,\nSylvain\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hello,

Given a complete manifold X, Let d be the de Rham operator and A a 1
form (with value in R). The magnetic laplacian is defined as
$\Delta_A=(d+iA)*(d+iA)$.

In the general case, let be E be vector bundle on X. Let $\nabla$ be a
connection on E. I define the "Yang-Mills Laplacian" as being
$\Delta_YM=\nabla*\nabla$.

Let E be associated to a principal bundle with group G, i.e. E is
locally XxF where G acts on F.

1. Is the "Yang-Mills Laplacian" of physical interest? (I guess so
because it generalizes the magnetic laplacian)

2. Are there special G and representation F of particular interest?

3. If the cohomology group $H^1$ of X is trivial, we have gauge
invariance for the magnetic laplacian, i.e. it depends, up to unitary
equivalence, only on the curvature dA of the connection. What is the
analogous property for the YM Laplacian ?

Sylvain

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On $2005-06-02,$ sylvain.golenia@gmail.com wrote: > Hello, > > Given a complete manifold X, Let d be the de Rham operator and A a 1 > form (with value in R). The magnetic laplacian is defined as > $\Delta_A=(d+iA)*(d+iA)$. One of the i's should be $a -i$. Hermicity. > In the general case, let be E be vector bundle on X. Let $\nabla$ be a > connection on E. I define the "Yang-Mills Laplacian" as being > $\Delta_YM=\nabla*\nabla$. > > Let E be associated to a principal bundle with group G, i.e. E is > locally XxF where G acts on F. > > 1. Is the "Yang-Mills Laplacian" of physical interest? (I guess so > because it generalizes the magnetic laplacian) Yes. Since you are using the therm Yang-Mills, I presume you already know how to use this Laplacian to form the Yang-Mills Lagrangian. Then you question can be interpreted as: Is the Yang-Mills Lagrangian used to describe any physical systems? The answer is Yes. The system in question is the Standard Model of particle physics. Describes all known species of matter and all interactions except for gravity. > 2. Are there special G and representation F of particular interest? Yes. The gauge group for the Standard Model is U(1)xSU(2)xSU(3). Roughly, they correspond to electromagnetic, weak, and strong interactions. There is also a specific choice of representation for the matter and radiation fields. The gauge bosons live in the adjoint representation, while matter fields live (mostly I think) in the fundamental representations of the gauge group with respect to whichy they carry charge (electromangnetic, weak, or strong charge). Perhaps someone more knowledgeable can fill in the details or cite a good reference for precisely which representations are used. > 3. If the cohomology group $H^1$ of X is trivial, we have gauge > invariance for the magnetic laplacian, i.e. it depends, up to unitary > equivalence, only on the curvature dA of the connection. What is the > analogous property for the YM Laplacian ? It is invariant under the choice of trivialization of the fibre bundle. Igor



Hi, Thanks for the reply! Igor Khavkine wrote: > On $2005-06-02,$ sylvain.golenia@gmail.com wrote: > > In the general case, let be E be vector bundle on X. Let $\nabla$ be a > > connection on E. I define the "Yang-Mills Laplacian" as being > > $\Delta_YM=\nabla*\nabla$. > > > > Let E be associated to a principal bundle with group G, i.e. E is > > locally XxF where G acts on F. > > > > 1. Is the "Yang-Mills Laplacian" of physical interest? (I guess so > > because it generalizes the magnetic laplacian) > > Yes. Since you are using the therm Yang-Mills, I presume you already > know how to use this Laplacian to form the Yang-Mills Lagrangian. Then > you question can be interpreted as: Is the Yang-Mills Lagrangian used to > describe any physical systems? I am not sure to know how to use this Laplacian to form the Yang-Mills Lagrangian. Maybe it is naive but it is interesting in itself to study the evolution of a particle under the action of the Yang-Mills Laplacian, i.e. to study the spectral properties of $\Delta_YM$ and to understand $\exp(-i t \Delta_YM)f$ for a f belonging to the spectral projections of $\Delta_YM ?$ Thanks, Sylvain

## Yang-Mills Fields

<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>On 2005-06-03, sylvain.golenia@gmail.com &lt;sylvain.golenia@gmail.com&gt; wrote:\n&gt; Igor Khavkine wrote:\n&gt;&gt; On 2005-06-02, sylvain.golenia@gmail.com &lt;sylvain.golenia@gmail.com&gt; wrote:\n&gt;\n&gt;&gt; &gt; In the general case, let be E be vector bundle on X. Let \\nabla be a\n&gt;&gt; &gt; connection on E. I define the "Yang-Mills Laplacian" as being\n&gt;&gt; &gt; \\Delta_YM=\\nabla*\\nabla.\n&gt;&gt; &gt;\n&gt;&gt; &gt; Let E be associated to a principal bundle with group G, i.e. E is\n&gt;&gt; &gt; locally XxF where G acts on F.\n&gt;&gt; &gt;\n&gt;&gt; &gt; 1. Is the "Yang-Mills Laplacian" of physical interest? (I guess so\n&gt;&gt; &gt; because it generalizes the magnetic laplacian)\n&gt;&gt;\n&gt;&gt; Yes. Since you are using the therm Yang-Mills, I presume you already\n&gt;&gt; know how to use this Laplacian to form the Yang-Mills Lagrangian. Then\n&gt;&gt; you question can be interpreted as: Is the Yang-Mills Lagrangian used to\n&gt;&gt; describe any physical systems?\n&gt;\n&gt; I am not sure to know how to use this Laplacian to form the Yang-Mills\n&gt; Lagrangian.\n&gt;\n&gt; Maybe it is naive but it is interesting in itself to study the\n&gt; evolution of a particle under the action of the Yang-Mills Laplacian,\n&gt; i.e. to study the spectral properties of \\Delta_YM and to understand\n&gt; exp(-i t \\Delta_YM)f for a f belonging to the spectral projections of\n&gt; \\Delta_YM ?\n\nThe YM Lagrangian is written simply as\n\nL = psi^* D_YM psi or L = phi^* Delta_YM phi\n\nwhere D_YM is the YM covariant derivative and Delta_YM is the\ncorresponding Laplacian. psi is a Lorentz spinor field valued in a\nrepresentation of the group G, and phi is a Lorentz scalar field valued\nin a representation of G. These define field theories (infinitely many\ndegrees of freedom), since the fields psi are the dynamical variables.\n\nThat\'s what is usually done. What you seem to be suggesting is writing\ndown a single particle theory (finitely many degrees of freedom) with\nthe Hamiltonian\n\nH = -Delta_YM.\n\nYou can do that. You just have to notice that Delta_YM is no longer a\nLaplacian on space-time, but instead a Laplacian on spacial slices of\nyour spacetime. You have to pull the connection back onto the spacial\nslices to define it.\n\nIf the group G is U(1), you obtain the non-relativistic equations of\nmotion of a quantum particle in an external magnetic field. If you\nreplace G by another group, the situation is analogous.\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>On $2005-06-03,$ sylvain.golenia@gmail.com <sylvain.golenia@gmail.com> wrote:
> Igor Khavkine wrote:
>> On $2005-06-02,$ sylvain.golenia@gmail.com <sylvain.golenia@gmail.com> wrote:

>
>> > In the general case, let be E be vector bundle on X. Let $\nabla$ be a
>> > connection on E. I define the "Yang-Mills Laplacian" as being

>> $> \Delta_YM=\nabla*\nabla$.
>> >
>> > Let E be associated to a principal bundle with group G, i.e. E is
>> > locally XxF where G acts on F.
>> >
>> > 1. Is the "Yang-Mills Laplacian" of physical interest? (I guess so
>> > because it generalizes the magnetic laplacian)

>>
>> Yes. Since you are using the therm Yang-Mills, I presume you already
>> know how to use this Laplacian to form the Yang-Mills Lagrangian. Then
>> you question can be interpreted as: Is the Yang-Mills Lagrangian used to
>> describe any physical systems?

>
> I am not sure to know how to use this Laplacian to form the Yang-Mills
> Lagrangian.
>
> Maybe it is naive but it is interesting in itself to study the
> evolution of a particle under the action of the Yang-Mills Laplacian,
> i.e. to study the spectral properties of $\Delta_YM$ and to understand
> $\exp(-i t \Delta_YM)f$ for a f belonging to the spectral projections of
> $\Delta_YM ?$

The YM Lagrangian is written simply as

$$L = \psi^* D_{YM} \psi[/itex] or $L = \phi^* \Delta_YM \phi$$ where [itex]D_{YM}$ is the YM covariant derivative and $\Delta_YM$ is the
corresponding Laplacian$. \psi$ is a Lorentz spinor field valued in a
representation of the group G, and $\phi$ is a Lorentz scalar field valued
in a representation of G. These define field theories (infinitely many
degrees of freedom), since the fields $\psi$ are the dynamical variables.

That's what is usually done. What you seem to be suggesting is writing
down a single particle theory (finitely many degrees of freedom) with
the Hamiltonian

$H = -\Delta_YM$.

You can do that. You just have to notice that $\Delta_YM$ is no longer a
Laplacian on space-time, but instead a Laplacian on spacial slices of
your spacetime. You have to pull the connection back onto the spacial
slices to define it.

If the group G is U(1), you obtain the non-relativistic equations of
motion of a quantum particle in an external magnetic field. If you
replace G by another group, the situation is analogous.

Hope this helps.

Igor

 Hi, Thanks for the reply! Igor Khavkine wrote: > On 2005-06-02, sylvain.golenia@gmail.com wrote: > > In the general case, let be E be vector bundle on X. Let \nabla be a > > connection on E. I define the "Yang-Mills Laplacian" as being > > \Delta_YM=\nabla*\nabla. > > > > Let E be associated to a principal bundle with group G, i.e. E is > > locally XxF where G acts on F. > > > > 1. Is the "Yang-Mills Laplacian" of physical interest? (I guess so > > because it generalizes the magnetic laplacian) > > Yes. Since you are using the therm Yang-Mills, I presume you already > know how to use this Laplacian to form the Yang-Mills Lagrangian. Then > you question can be interpreted as: Is the Yang-Mills Lagrangian used to > describe any physical systems? I am not sure to know how to use this Laplacian to form the Yang-Mills Lagrangian. Maybe it is naive but it is interesting in itself to study the evolution of a particle under the action of the Yang-Mills Laplacian, i.e. to study the spectral properties of \Delta_YM and to understand exp(-i t \Delta_YM)f for a f belonging to the spectral projections of \Delta_YM ? Thanks, Sylvain