Defining a non-standard algebraic number

Question

I need to perform matrix computations, where elements of matrices not only contain $i$ (imaginary unit), but also a specific "unit" number, let call it $c$, with non-standard algebraic properties: $c\cdot c=1$, but $c \cdot i = -i \cdot c$, and commutation with real numbers is trivial. That is some algebra with user-defined unit elements $(1,i,c)$ and certain relations between them (perhaps, a parody on quaternions).

(in fact, this is the "operator" of complex conjugation, which enters matrix elements, and in quantum field theory-related problems will be acting afterwards on whatever stands right of it; but for now we need to manipulate it like a number, but oddly commuting with I).

How can one tell Mathematica to handle this new variable $c$ with such properties? Maybe, there is an embedded object with such properties (like there are embedded quaternions and packages to deal with Dirac matrices)? In general, how one can define own algebra with arbitrary properties?

Answer 1

You might try embedding into a larger matrix in a way that the relations work out. So the imaginary unit becomes {{0,1},{1,0}}, real values become diagonal 2x2 matrices, and for this particular c it looks like {{1,0},{0,-1}} will work. This will involve simple replacements, some array flattening and unflattening, and a simple undoing of the replacements. Also important is to realize that we really have four generators, not 3, insofar as resulting 2x2 matrices will need to be reconstructed as a real part, an imaginary part, a c part, and one other reverse diagonal which can be in terms of either "I*c" or "c*I". Since multiplications is commutative we cannot use these explicitly and so I will just use a surrogate variable, called ci, for the latter. Also I will use i from the beginning to represent out imaginary unit, rather than Complex[0,1]. So here is how we proceed.

Start with replacements that take us from scalar to 2x2 matrix form, and reverse.

replacements = {i -> {{0, 1}, {1, 0}}, c -> {{1, 0}, {0, -1}}};

cirForm[mat_?MatrixQ /; 
   Dimensions[mat] === {2, 2}] := (mat[[1, 1]] + mat[[2, 2]])/2 + 
  c*(mat[[1, 1]] - mat[[2, 2]])/2 + i*(mat[[1, 2]] + mat[[2, 1]])/2 + 
  ci*(mat[[1, 2]] - mat[[2, 1]])/2

We create an example using two 4x4 matrices containing real, imaginary, and c parts.

SeedRandom[1234];
rand1 = RandomInteger[{-10, 10}, {3, 4, 4}];
rand2 = RandomInteger[{-10, 10}, {3, 4, 4}];
mat1 = {1, i, c}.rand1
mat2 = {1, i, c}.rand2

(* Out[1606]= {{-9 - 6 c + 6 i, 10 - 8 c + 8 i, 
  10 + 6 c + 10 i, -9 - 6 c - 3 i}, {-10 + 6 c + 4 i, 6 - 9 c + 6 i, 
  7 + 3 c - 4 i, -6 + 7 c - 9 i}, {-5 - c + 5 i, 9 + 10 c - 3 i, 
  5 + c - 2 i, 6 - 4 c + 3 i}, {9 + 4 c + 3 i, -8 + 8 c - 9 i, 
  2 + c + 5 i, -5 - 8 c - 2 i}}

Out[1607]= {{-6 - 8 c - 5 i, 6 - 6 c - i, 5 - 8 c + 6 i, 
  1 + 10 i}, {-1 - 2 i, 4 + 6 c - 10 i, 2 + 4 c + 7 i, 
  9 - 6 c - i}, {9 + 2 c + 8 i, 10 + c - 9 i, 
  1 + 9 c - 8 i, -9 - 9 c + 4 i}, {-2 + 4 c - 5 i, 
  7 + 8 i, -8 + 4 c - 2 i, -8 + c + 2 i}} *)

Now flatten into 8x8s.

mat1Full = ArrayFlatten[mat1 /. replacements, 2]
mat2Full = ArrayFlatten[mat2 /. replacements, 2]

(* Out[1608]= {{-15, -3, 2, 18, 16, 20, -15, -12}, {-3, -3, 18, 18, 20, 
  4, -12, -3}, {-4, -6, -3, 12, 10, 3, 1, -15}, {-6, -16, 12, 15, 3, 
  4, -15, -13}, {-6, 0, 19, 6, 6, 3, 2, 9}, {0, -4, 6, -1, 3, 4, 9, 
  10}, {13, 12, 0, -17, 3, 7, -13, -7}, {12, 5, -17, -16, 7, 1, -7, 3}}

Out[1609]= {{-14, -11, 0, 5, -3, 11, 1, 11}, {-11, 2, 5, 12, 11, 13, 
  11, 1}, {-1, -3, 10, -6, 6, 9, 3, 8}, {-3, -1, -6, -2, 9, -2, 8, 
  15}, {11, 17, 11, 1, 10, -7, -18, -5}, {17, 7, 1, 9, -7, -8, -5, 
  0}, {2, -7, 7, 15, -4, -10, -7, -6}, {-7, -6, 15, 
  7, -10, -12, -6, -9}} *)

Take the product and repartition into a 4x4 comprised of 2x2s.

matProductFull = mat1Full.mat2Full

(* Out[1600]= {{757, 724, -192, -272, 386, -200, -109, 236}, {288, 425, 
  152, -340, 496, 38, -116, 377}, {357, 303, -237, -151, 
  261, -97, -95, 185}, {365, 245, -313, -601, 241, 57, 83, 431}, {105,
   58, 372, -30, 127, -101, -92, 53}, {90, -69, 296, 
  162, -151, -259, -231, -130}, {-88, 131, 6, 65, 43, 470, 53, 
  26}, {-99, 102, 25, 186, -166, 53, -212, -259}} *)

matProduct = Partition[matProductFull, {2, 2}]

(* Out[1601]= {{{{757, 724}, {288, 
    425}}, {{-192, -272}, {152, -340}}, {{386, -200}, {496, 
    38}}, {{-109, 236}, {-116, 377}}}, {{{357, 303}, {365, 
    245}}, {{-237, -151}, {-313, -601}}, {{261, -97}, {241, 
    57}}, {{-95, 185}, {83, 431}}}, {{{105, 
    58}, {90, -69}}, {{372, -30}, {296, 
    162}}, {{127, -101}, {-151, -259}}, {{-92, 
    53}, {-231, -130}}}, {{{-88, 131}, {-99, 102}}, {{6, 65}, {25, 
    186}}, {{43, 470}, {-166, 53}}, {{53, 26}, {-212, -259}}}} *)

Rewrite in "cir" form.

cirmatProduct = Map[cirForm, matProduct, {2}]

(* Out[1602]= {{591 + 166 c + 218 ci + 506 i, -266 + 74 c - 212 ci - 
   60 i, 212 + 174 c - 348 ci + 148 i, 
  134 - 243 c + 176 ci + 60 i}, {301 + 56 c - 31 ci + 334 i, -419 + 
   182 c + 81 ci - 232 i, 159 + 102 c - 169 ci + 72 i, 
  168 - 263 c + 51 ci + 134 i}, {18 + 87 c - 16 ci + 74 i, 
  267 + 105 c - 163 ci + 133 i, -66 + 193 c + 25 ci - 126 i, -111 + 
   19 c + 142 ci - 89 i}, {7 - 95 c + 115 ci + 16 i, 
  96 - 90 c + 20 ci + 45 i, 
  48 - 5 c + 318 ci + 152 i, -103 + 156 c + 119 ci - 93 i}} *)

Answer 2

Would this work? I am using a special symbol for your $c$, to help distinguish it from any other variable:

Clear[\[ConstantC]]
\[ConstantC] /: \[ConstantC] \[ConstantC] = 1
\[ConstantC] /: \[ConstantC] I = -I \[ConstantC]

This looks like the following in a notebook:

With these definitions, I can evaluate:

and obtain: