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<h1 class="chapter"> 29. atensor </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC132">29.1 Introduction to atensor</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC133">29.2 Functions and Variables for atensor</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
</table>

<p><a name="Item_003a-Introduction-to-atensor"></a>
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<a name="Introduction-to-atensor"></a>
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<h2 class="section"> 29.1 Introduction to atensor </h2>

<p><code>atensor</code> is an algebraic tensor manipulation package. To use <code>atensor</code>,
type <code>load(atensor)</code>, followed by a call to the <code>init_atensor</code>
function.
</p>
<p>The essence of <code>atensor</code> is a set of simplification rules for the
noncommutative (dot) product operator (&quot;<code>.</code>&quot;). <code>atensor</code> recognizes
several algebra types; the corresponding simplification rules are put
into effect when the <code>init_atensor</code> function is called.
</p>
<p>The capabilities of <code>atensor</code> can be demonstrated by defining the
algebra of quaternions as a Clifford-algebra Cl(0,2) with two basis
vectors. The three quaternionic imaginary units are then the two
basis vectors and their product, i.e.:
</p>
<pre class="example">    i = v     j = v     k = v  . v
         1         2         1    2
</pre>
<p>Although the <code>atensor</code> package has a built-in definition for the
quaternion algebra, it is not used in this example, in which we
endeavour to build the quaternion multiplication table as a matrix:
</p>
<pre class="example">
(%i1) load(atensor);
(%o1)       /share/tensor/atensor.mac
(%i2) init_atensor(clifford,0,0,2);
(%o2)                                done
(%i3) atensimp(v[1].v[1]);
(%o3)                                 - 1
(%i4) atensimp((v[1].v[2]).(v[1].v[2]));
(%o4)                                 - 1
(%i5) q:zeromatrix(4,4);
                                [ 0  0  0  0 ]
                                [            ]
                                [ 0  0  0  0 ]
(%o5)                           [            ]
                                [ 0  0  0  0 ]
                                [            ]
                                [ 0  0  0  0 ]
(%i6) q[1,1]:1;
(%o6)                                  1
(%i7) for i thru adim do q[1,i+1]:q[i+1,1]:v[i];
(%o7)                                done
(%i8) q[1,4]:q[4,1]:v[1].v[2];
(%o8)                               v  . v
                                     1    2
(%i9) for i from 2 thru 4 do for j from 2 thru 4 do
      q[i,j]:atensimp(q[i,1].q[1,j]);
(%o9)                                done
(%i10) q;
                   [    1        v         v      v  . v  ]
                   [              1         2      1    2 ]
                   [                                      ]
                   [   v         - 1     v  . v    - v    ]
                   [    1                 1    2      2   ]
(%o10)             [                                      ]
                   [   v      - v  . v     - 1      v     ]
                   [    2        1    2              1    ]
                   [                                      ]
                   [ v  . v      v        - v       - 1   ]
                   [  1    2      2          1            ]
</pre>
<p><code>atensor</code> recognizes as base vectors indexed symbols, where the symbol
is that stored in <code>asymbol</code> and the index runs between 1 and <code>adim</code>.
For indexed symbols, and indexed symbols only, the bilinear forms
<code>sf</code>, <code>af</code>, and <code>av</code> are evaluated. The evaluation
substitutes the value of <code>aform[i,j]</code> in place of <code>fun(v[i],v[j])</code>
where <code>v</code> represents the value of <code>asymbol</code> and <code>fun</code> is
either <code>af</code> or <code>sf</code>; or, it substitutes <code>v[aform[i,j]]</code>
in place of <code>av(v[i],v[j])</code>.
</p>
<p>Needless to say, the functions <code>sf</code>, <code>af</code> and <code>av</code>
can be redefined.
</p>
<p>When the <code>atensor</code> package is loaded, the following flags are set:
</p>
<pre class="example">dotscrules:true;
dotdistrib:true;
dotexptsimp:false;
</pre>
<p>If you wish to experiment with a nonassociative algebra, you may also
consider setting <code>dotassoc</code> to <code>false</code>. In this case, however,
<code>atensimp</code> will not always be able to obtain the desired
simplifications.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Tensors">Tensors</a>
 &middot;
<a href="maxima_95.html#Category_003a-Share-packages">Share packages</a>
 &middot;
<a href="maxima_95.html#Category_003a-Package-atensor">Package atensor</a>
</p>
</div>


<p><a name="Item_003a-Functions-and-Variables-for-atensor"></a>
</p><hr size="6">
<a name="Functions-and-Variables-for-atensor"></a>
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<a name="SEC133"></a>
<h2 class="section"> 29.2 Functions and Variables for atensor </h2>

<p><a name="Item_003a-init_005fatensor"></a>
</p><dl>
<dt><u>Function:</u> <b>init_atensor</b><i> (<var>alg_type</var>, <var>opt_dims</var>)</i>
<a name="IDX1095"></a>
</dt>
<dt><u>Function:</u> <b>init_atensor</b><i> (<var>alg_type</var>)</i>
<a name="IDX1096"></a>
</dt>
<dd><p>Initializes the <code>atensor</code> package with the specified algebra type. <var>alg_type</var>
can be one of the following:
</p>
<p><code>universal</code>: The universal algebra has no commutation rules.
</p>
<p><code>grassmann</code>: The Grassman algebra is defined by the commutation
relation <code>u.v+v.u=0</code>.
</p>
<p><code>clifford</code>: The Clifford algebra is defined by the commutation
relation <code>u.v+v.u=-2*sf(u,v)</code> where <code>sf</code> is a symmetric
scalar-valued function. For this algebra, <var>opt_dims</var> can be up
to three nonnegative integers, representing the number of positive,
degenerate, and negative dimensions of the algebra, respectively. If
any <var>opt_dims</var> values are supplied, <code>atensor</code> will configure the
values of <code>adim</code> and <code>aform</code> appropriately. Otherwise,
<code>adim</code> will default to 0 and <code>aform</code> will not be defined.
</p>
<p><code>symmetric</code>: The symmetric algebra is defined by the commutation
relation <code>u.v-v.u=0</code>.
</p>
<p><code>symplectic</code>: The symplectic algebra is defined by the commutation
relation <code>u.v-v.u=2*af(u,v)</code> where <code>af</code> is an antisymmetric
scalar-valued function. For the symplectic algebra, <var>opt_dims</var> can
be up to two nonnegative integers, representing the nondegenerate and
degenerate dimensions, respectively. If any <var>opt_dims</var> values are
supplied, <code>atensor</code> will configure the values of <code>adim</code> and <code>aform</code>
appropriately. Otherwise, <code>adim</code> will default to 0 and <code>aform</code>
will not be defined.
</p>
<p><code>lie_envelop</code>: The algebra of the Lie envelope is defined by the
commutation relation <code>u.v-v.u=2*av(u,v)</code> where <code>av</code> is
an antisymmetric function.
</p>
<p>The <code>init_atensor</code> function also recognizes several predefined
algebra types:
</p>
<p><code>complex</code> implements the algebra of complex numbers as the
Clifford algebra Cl(0,1). The call <code>init_atensor(complex)</code> is
equivalent to <code>init_atensor(clifford,0,0,1)</code>.
</p>
<p><code>quaternion</code> implements the algebra of quaternions. The call
<code>init_atensor(quaternion)</code> is equivalent to
<code>init_atensor(clifford,0,0,2)</code>.
</p>
<p><code>pauli</code> implements the algebra of Pauli-spinors as the Clifford-algebra
Cl(3,0). A call to <code>init_atensor(pauli)</code> is equivalent to
<code>init_atensor(clifford,3)</code>.
</p>
<p><code>dirac</code> implements the algebra of Dirac-spinors as the Clifford-algebra
Cl(3,1). A call to <code>init_atensor(dirac)</code> is equivalent to
<code>init_atensor(clifford,3,0,1)</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-atensor">Package atensor</a>
</p>
</div>


</dd></dl>


<p><a name="Item_003a-atensimp"></a>
</p><dl>
<dt><u>Function:</u> <b>atensimp</b><i> (<var>expr</var>)</i>
<a name="IDX1097"></a>
</dt>
<dd><p>Simplifies an algebraic tensor expression <var>expr</var> according to the rules
configured by a call to <code>init_atensor</code>. Simplification includes
recursive application of commutation relations and resolving calls
to <code>sf</code>, <code>af</code>, and <code>av</code> where applicable. A
safeguard is used to ensure that the function always terminates, even
for complex expressions.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-atensor">Package atensor</a>
 &middot;
<a href="maxima_95.html#Category_003a-Simplification-functions">Simplification functions</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-alg_005ftype"></a>
</p><dl>
<dt><u>Function:</u> <b>alg_type</b>
<a name="IDX1098"></a>
</dt>
<dd><p>The algebra type. Valid values are <code>universal</code>, <code>grassmann</code>,
<code>clifford</code>, <code>symmetric</code>, <code>symplectic</code> and <code>lie_envelop</code>.
</p>
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</dd></dl>

<p><a name="Item_003a-adim"></a>
</p><dl>
<dt><u>Variable:</u> <b>adim</b>
<a name="IDX1099"></a>
</dt>
<dd><p>Default value: 0
</p>
<p>The dimensionality of the algebra. <code>atensor</code> uses the value of <code>adim</code>
to determine if an indexed object is a valid base vector.  See <code>abasep</code>.
</p>
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<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-atensor">Package atensor</a>
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</dd></dl>

<p><a name="Item_003a-aform"></a>
</p><dl>
<dt><u>Variable:</u> <b>aform</b>
<a name="IDX1100"></a>
</dt>
<dd><p>Default value: <code>ident(3)</code>
</p>
<p>Default values for the bilinear forms <code>sf</code>, <code>af</code>, and
<code>av</code>. The default is the identity matrix <code>ident(3)</code>.
</p>
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<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-atensor">Package atensor</a>
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</div>


</dd></dl>

<p><a name="Item_003a-asymbol"></a>
</p><dl>
<dt><u>Variable:</u> <b>asymbol</b>
<a name="IDX1101"></a>
</dt>
<dd><p>Default value: <code>v</code>
</p>
<p>The symbol for base vectors.
</p>
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</dd></dl>

<p><a name="Item_003a-sf"></a>
</p><dl>
<dt><u>Function:</u> <b>sf</b><i> (<var>u</var>, <var>v</var>)</i>
<a name="IDX1102"></a>
</dt>
<dd><p>A symmetric scalar function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using <code>abasep</code> and if that is the case, substitutes the
corresponding value from the matrix <code>aform</code>.
</p>
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</p>
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</dd></dl>

<p><a name="Item_003a-af"></a>
</p><dl>
<dt><u>Function:</u> <b>af</b><i> (<var>u</var>, <var>v</var>)</i>
<a name="IDX1103"></a>
</dt>
<dd><p>An antisymmetric scalar function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using <code>abasep</code> and if that is the case, substitutes the
corresponding value from the matrix <code>aform</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-atensor">Package atensor</a>
</p>
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</dd></dl>

<p><a name="Item_003a-av"></a>
</p><dl>
<dt><u>Function:</u> <b>av</b><i> (<var>u</var>, <var>v</var>)</i>
<a name="IDX1104"></a>
</dt>
<dd><p>An antisymmetric function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using <code>abasep</code> and if that is the case, substitutes the
corresponding value from the matrix <code>aform</code>.
</p>
<p>For instance:
</p>
<pre class="example">(%i1) load(atensor);
(%o1)       /share/tensor/atensor.mac
(%i2) adim:3;
(%o2)                                  3
(%i3) aform:matrix([0,3,-2],[-3,0,1],[2,-1,0]);
                               [  0    3   - 2 ]
                               [               ]
(%o3)                          [ - 3   0    1  ]
                               [               ]
                               [  2   - 1   0  ]
(%i4) asymbol:x;
(%o4)                                  x
(%i5) av(x[1],x[2]);
(%o5)                                 x
                                       3
</pre>
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<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-atensor">Package atensor</a>
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</dd></dl>


<p><a name="Item_003a-abasep"></a>
</p><dl>
<dt><u>Function:</u> <b>abasep</b><i> (<var>v</var>)</i>
<a name="IDX1105"></a>
</dt>
<dd><p>Checks if its argument is an <code>atensor</code> base vector. That is, if it is
an indexed symbol, with the symbol being the same as the value of
<code>asymbol</code>, and the index having a numeric value between 1
and <code>adim</code>.
</p>
<div class=categorybox>


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 &middot;
<a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
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