RE: Defining anti-symmetric operation. New ideas requested.

• To: mathgroup at smc.vnet.net
• Subject: [mg47313] RE: [mg47223] Defining anti-symmetric operation. New ideas requested.
• From: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com>
• Date: Mon, 5 Apr 2004 05:22:50 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```By an error of mine, this did not reach MathGroup. The basic idea is the
same as communicated by Daniel Lichtblau (and sketched by Jens-Peer Kuska),
namely to convert any expression with Commute to a "normal form", which
contains the definition (or the default). This then avoids infinite
recursion. Here below a slightly polished version of my attempt.

>-----Original Message-----
>From: Oleksandr Pavlyk [mailto:pavlyk at phys.psu.edu]
To: mathgroup at smc.vnet.net
>Sent: Wednesday, March 31, 2004 9:58 AM
>To: mathgroup at smc.vnet.net
>Subject: [mg47313] [mg47223] Defining anti-symmetric operation. New ideas
>requested.
>
>
>Dear MathGroup,
>
>In order to explain the problem I would need to
>say few words about what I am trying to accomplish.
>
>I am often define bare bones of a Lie algebra,
>and check its consistency by Mathematica. To that
>end I have some generators, and I define their
>commutation relation. As an example let us take
>SU(2) with 3 generators  H[], X[], Y[]
>
>To define the algebra I write
>
>    Commute[ H[], X[] ] = Y[]
>    Commute[ X[], Y[] ] = H[]
>    Commute[ Y[], H[] ] = X[]
>
>now because of properties of Commute,
>
>Commute[ OperatorA, OperatorB ] = - Commute[OperatorB, OperatorA]
>
>So far I am realizing this property by defining
>
>    Commute[ X[], H[] ] := -Commute[ H[], X[] ]
>    Commute[ Y[], X[] ] := -Commute[ X[], Y[] ]
>    Commute[ H[], Y[] ] := -Commute[ Y[], H[] ]
>
>So I come to my question. I would like Mathematica to
>define automatically
>
>    Commute[OperatorB, OperatorA]
>
>every time I define Commute[OperatorA, OperatorB].
>
>I tried to approach the problem by examining
>DownValues of Commute.
>
>(* This is evaluated if no direct definition was matched *)
>Commute[a_, b_]:=Module[{res},
>   res = HoldComplete[Commute[b,a]]/.DownValues[Commute];
>   If[ res === HoldComplete[Commute[b,a]], (* If nothing in
>DownValues *)
>         Return[0]      (* we declare everything else commuting *)
>       ,
>         Return[ -ReleaseHold[res] ]
>   ]
>]
>
>This definition indeed gives Commute[ X[], H[] ] = -Y[], but if I try
>it on some undefined generators
>
>   Commute[ J[], H[] ]
>
>I get infinite recursion, which is there because the very definition
>of trick solving part of Commute is also in DownValues, and
>hence we get
>
>  res = HoldComplete[ Module[{},.... ] ]
>
>Any ideas on how to encode anti-symmetric properties of Commute would
>be very appreciated.
>
>I am sorry for slightly long post. The Mathematica notebook with the
>code is posted at
>
>    http://www.pavlyk.com/SU(2).nb
>
>Thank you,
>Sasha
>

An idea:

define the anticommuting property with respect to normal order:

In[2]:= Commute[a__] := Signature[{a}]Sort[Unevaluated[Commute[a]]]

If there is no definition for normal order, things commute:

In[3]:= Commute[a__] /; OrderedQ[{a}] := 0

You now may work, but have to adhere to a strict discipline as to set up
definitions only in normal order. This certainly is inconvenient. Else we
might try to convert to normal order at definiton time:

In[4]:=
Commute /: Set[Commute[a__], c_] /; ! OrderedQ[{a}] :=
With[{normalOrderedArg = Sequence @@ Sort[{a}]},
Commute[normalOrderedArg] = Signature[{a}] c]

Now your definitions: Define commutation relations for SU(2)

In[5]:= Commute[H[], X[]] = Y[];

In[6]:= Commute[X[], Y[]] = H[];

In[7]:= Commute[Y[], H[]] = X[];

The last one is not given in normal order, let's see what happens:

In[8]:= Commute[H[], X[]]
Out[8]= Y[]

In[9]:= Commute[X[], H[]]
Out[9]= -Y[]

In[10]:= Commute[X[], Y[]]
Out[10]= H[]

In[11]:= Commute[Y[], X[]]
Out[11]= -H[]

In[12]:= Commute[Y[], H[]]
Out[12]= X[]

In[13]:= Commute[H[], Y[]]
Out[13]= -X[]

In[14]:= Commute[J[], H[]]
Out[14]= 0

It appears to work; but what had been defined effectively?

In[16]:= ?Commute
>From In[16]:=
Global`Commute
>From In[16]:=
(Commute[a__] = c_) /; ! OrderedQ[{a}] ^:=
With[{normalOrderedArg = Sequence @@ Sort[{a}]},
Commute[normalOrderedArg] = Signature[{a}] c]

Commute[H[], X[]] = Y[]

Commute[H[], Y[]] = -X[]

Commute[X[], Y[]] = H[]

Commute[a__] /; OrderedQ[{a}] := 0

Commute[a__] := Signature[{a}] Sort[Unevaluated[Commute[a]]]

If you want to to define Commute only for two arguments, just replace
everywhere Commute[a__] by Commute[a_,b_], and {a} by {a,b}, Commute[a] by
Commute[a,b].

--
Hartmut Wolf

```

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