\chapter[PHYSOP: Operator Calculus]%
{PHYSOP: Operator calculus in quantum theory}
\label{PHYSOP}
\typeout{{PHYSOP: Operator calculus in quantum theory}}
{\footnotesize
\begin{center}
Mathias Warns \\
Physikalisches Institut der Universit\"at Bonn \\
Endenicher Allee 11--13 \\
D--5300 BONN 1, Germany \\[0.05in]
e--mail: UNP008@DBNRHRZ1.bitnet
\end{center}
}
\ttindex{PHYSOP}
The package PHYSOP has been designed to meet the requirements of
theoretical physicists looking for a
computer algebra tool to perform complicated calculations
in quantum theory
with expressions containing operators. These operations
consist mainly in the calculation of commutators between operator
expressions and in the evaluations of operator matrix elements
in some abstract space.
\section{The NONCOM2 Package}
The package NONCOM2 redefines some standard \REDUCE\ routines
in order to modify the way noncommutative operators are handled by the
system. It redefines the \f{NONCOM}\ttindex{NONCOM} statement in
a way more suitable for calculations in physics. Operators have now to
be declared noncommutative pairwise, {\em i.e.\ }coding: \\
\begin{verbatim}
NONCOM A,B;
\end{verbatim}
declares the operators \f{A} and \f{B} to be noncommutative but allows them
to commute with any other (noncommutative or not) operator present in
the expression. In a similar way if one wants {\em e.g.\ }\f{A(X)} and
\f{A(Y)} not to commute, one has now to code:
\begin{verbatim}
NONCOM A,A;
\end{verbatim}
A final example should make
the use of the redefined \f{NONCOM} statement clear:
\begin{verbatim}
NONCOM A,B,C;
\end{verbatim}
declares \f{A} to be noncommutative with \f{B} and \f{C},
\f{B} to be noncommutative
with \f{A} and \f{C} and \f{C} to be noncommutative
with \f{A} and \f{B}.
Note that after these declaration
{\em e.g.\ }\f{A(X)} and \f{A(Y)}
are still commuting kernels.
Finally to keep the compatibility with standard \REDUCE\, declaring a
\underline{single} identifier using the \f{NONCOM} statement has the same
effect as in
standard \REDUCE.
From the user's point of view there are no other
new commands implemented by the package.
\section{The PHYSOP package}
The package PHYSOP implements a new \REDUCE\ data type to perform
calculations with physical operators. The noncommutativity of
operators is
implemented using the NONCOM2 package so this file should be loaded
prior to the use of PHYSOP.
\subsection{Type declaration commands}
The new \REDUCE\ data type PHYSOP implemented by the package allows the
definition of a new kind of operators ({\em i.e.\ }kernels carrying
an arbitrary
number of arguments). Throughout this manual, the name
``operator''
will refer, unless explicitly stated otherwise, to this new data type.
This data type is in turn
divided into 5 subtypes. For each of this subtype, a declaration command
has been defined:
\begin{description}
\item[\f{SCALOP A;} ]\ttindex{SCALOP} declares \f{A} to be a scalar
operator. This operator may
carry an arbitrary number of arguments; after the
declaration: \f{ SCALOP A; }
all kernels of the form
\f{A(J), A(1,N), A(N,L,M)}
are recognised by the system as being scalar operators.
\item[\f{VECOP V;} ]\ttindex{VECOP} declares \f{V} to be a vector operator.
As for scalar operators, the vector operators may carry an arbitrary
number of arguments. For example \f{V(3)} can be used to represent
the vector operator $\vec{V}_{3}$. Note that the dimension of space
in which this operator lives is \underline{arbitrary}.
One can however address a specific component of the
vector operator by using a special index declared as \f{PHYSINDEX} (see
below). This index must then be the first in the argument list
of the vector operator.
\item[\f{TENSOP C(3);} ] \ttindex{TENSOP}
declares \f{C} to be a tensor operator of rank 3. Tensor operators
of any fixed integer rank larger than 1 can be declared.
Again this operator may carry an arbitrary number of arguments
and the space dimension is not fixed.
The tensor
components can be addressed by using special \f{PHYSINDEX} indices
(see below) which have to be placed in front of all other
arguments in the argument list.
\item[\f{STATE U;} ]\ttindex{STATE} declares \f{U} to be a state, {\em i.e.\ }an
object on
which operators have a certain action. The state U can also carry an
arbitrary number of arguments.
\item[\f{PHYSINDEX X;} ]\ttindex{PHYSINDEX} declares \f{X} to be a special
index which will be used
to address components of vector and tensor operators.
\end{description}
A command \f{CLEARPHYSOP}\ttindex{CLEARPHYSOP}
removes
the PHYSOP type from an identifier in order to use it for subsequent
calculations. However it should be
remembered that \underline{no}
substitution rule is cleared by this function. It
is therefore left to the user's responsibility to clear previously all
substitution rules involving the identifier from which the PHYSOP type
is removed.
\subsection{Ordering of operators in an expression}
The ordering of kernels in an expression is performed according to
the following rules: \\
1. \underline{Scalars} are always ordered ahead of
PHYSOP operators in an expression.
The \REDUCE\ statement \f{KORDER}\ttindex{KORDER} can be used to control the
ordering of scalars but has no
effect on the ordering of operators.
2. The default ordering of operators follows the
order in which they have been declared (not the alphabetical one).
This ordering scheme can be changed using the command \f{OPORDER}.
\ttindex{OPORDER}
Its syntax is similar to the \f{KORDER} statement, {\em i.e.\ }coding:
\f{OPORDER A,V,F;}
means that all occurrences of the operator \f{A} are ordered ahead of
those of \f{V} etc. It is also possible to include operators
carrying
indices (both normal and special ones) in the argument list of
\f{OPORDER}. However including objects \underline{not}
defined as operators ({\em i.e.\ }scalars or indices) in the argument list
of the \f{OPORDER} command leads to an error.
3. Adjoint operators are placed by the declaration commands just
after the original operators on the \f{OPORDER} list. Changing the
place of an operator on this list means \underline{not} that the
adjoint operator is moved accordingly. This adjoint operator can
be moved freely by including it in the argument list of the
\f{OPORDER} command.
\subsection{Arithmetic operations on operators}
The following arithmetic operations are possible with
operator expressions: \\
1. Multiplication or division of an operator by a scalar.
2. Addition and subtraction of operators of the \underline{same}
type.
3. Multiplication of operators is only defined between two
\underline{scalar} operators.
4. The scalar product of two VECTOR operators is implemented
with a new function \f{DOT}\ttindex{DOT}. The system expands
the product of
two vector operators into an ordinary product of the components of these
operators by inserting a special index generated by the program.
To give an example, if one codes:
\begin{verbatim}
VECOP V,W;
V DOT W;
\end{verbatim}
the system will transform the product into:
\begin{verbatim}
V(IDX1) * W(IDX1)
\end{verbatim}
where \f{IDX1} is a \f{PHYSINDEX} generated by the system (called a
DUMMY INDEX in the following) to express the summation over the
components. The identifiers \f{IDXn} (\f{n} is a nonzero integer) are
reserved variables for this purpose and should not be used for other
applications. The arithmetic operator
\f{DOT} can be used both in infix and prefix form with two arguments.
5. Operators (but not states) can only be raised to an
\underline{integer} power. The system expands this power
expression into a product of the corresponding number of terms
inserting dummy indices if necessary. The following examples explain
the transformations occurring on power expressions (system output
is indicated with an \f{-->}):
\begin{verbatim}
SCALOP A; A**2;
--> A*A
VECOP V; V**4;
--> V(IDX1)*V(IDX1)*V(IDX2)*V(IDX2)
TENSOP C(2); C**2;
--> C(IDX3,IDX4)*C(IDX3,IDX4)
\end{verbatim}
Note in particular the way how the system interprets powers of
tensor operators which is different from the notation used in matrix
algebra.
6. Quotients of operators are only defined between
scalar operator expressions.
The system transforms the quotient of 2 scalar operators into the
product of the first operator times the inverse of the second one.
\begin{verbatim}
SCALOP A,B; A / B;
-1
A *( B )
\end{verbatim}
7. Combining the last 2 rules explains the way how the system
handles negative powers of operators:
\noindent
\begin{verbatim}
SCALOP B;
B**(-3);
-1 -1 -1
--> (B )*(B )*(B )
\end{verbatim}
The method of inserting dummy indices and expanding powers of
operators has been chosen to facilitate the handling of
complicated operator
expressions and particularly their application on states.
However it may be useful to get rid of these
dummy indices in order to enhance the readability of the
system's final output.
For this purpose the switch \f{CONTRACT}\ttindex{CONTRACT} has to
be turned on (\f{CONTRACT} is normally set to \f{OFF}).
The system in this case contracts over dummy indices reinserting the
\f{DOT} operator and reassembling the expanded powers. However due to
the predefined operator ordering the system may not remove all the
dummy indices introduced previously.
%%file).
\subsection{Special functions}
\subsubsection{Commutation relations}
If two PHYSOPs have been declared noncommutative using the (redefined)
\f{NONCOM}\ttindex{NONCOM} statement, it is possible to introduce in the environment
elementary (anti-) commutation relations between them. For
this purpose,
two scalar operators \f{COMM}\ttindex{COMM} and
\f{ANTICOMM}\ttindex{ANTICOMM} are available.
These operators are used in conjunction with \f{LET} statements.
Example:
\begin{verbatim}
SCALOP A,B,C,D;
LET COMM(A,B)=C;
FOR ALL N,M LET ANTICOMM(A(N),B(M))=D;
VECOP U,V,W; PHYSINDEX X,Y,Z;
FOR ALL X,Y LET COMM(V(X),W(Y))=U(Z);
\end{verbatim}
Note that if special indices are used as dummy variables in
\f{FOR ALL ... LET} constructs then these indices should have been
declared previously using the \f{PHYSINDEX} command.\ttindex{PHYSINDEX}
Every time the system
encounters a product term involving two
noncommutative operators which have to be reordered on account of the
given operator ordering, the list of available (anti-) commutators is
checked in the following way: First the system looks for a
\underline{commutation} relation which matches the product term. If it
fails then the defined \underline{anticommutation} relations are
checked. If there is no successful match the product term
\f{A*B} is replaced by: \\
\begin{verbatim}
A*B;
--> COMM(A,B) + B*A
\end{verbatim}
so that the user may introduce the commutation relation later on.
The user may want to force the system to look for
\underline{anticommutators} only; for this purpose a switch \f{ANTICOM}
\ttindex{ANTICOM}
is defined which has to be turned on ( \f{ANTICOM} is normally set to
\f{OFF}). In this case, the above example is replaced by:
\begin{verbatim}
ON ANTICOM;
A*B;
--> ANTICOMM(A,B) - B*A
\end{verbatim}
For the calculation of (anti-) commutators between complex
operator
expressions, the functions \f{COMMUTE}\ttindex{COMMUTE} and
\f{ANTICOMMUTE}\ttindex{ANTICOMMUTE} have been defined.
\begin{verbatim}
VECOP P,A,K;
PHYSINDEX X,Y;
FOR ALL X,Y LET COMM(P(X),A(Y))=K(X)*A(Y);
COMMUTE(P**2,P DOT A);
\end{verbatim}
\subsubsection{Adjoint expressions}
As has been already mentioned, for each operator and state defined
using the declaration commands, the system
generates automatically the corresponding adjoint operator. For the
calculation of the adjoint representation of a complicated
operator expression, a function \f{ADJ}\ttindex{ADJ} has been defined.
\begin{verbatim}
SCALOP A,B;
ADJ(A*B);
+ +
--> (A )*(B )
\end{verbatim}
\subsubsection{Application of operators on states}
A function \f{OPAPPLY}\ttindex{OPAPPLY} has been
defined for the application of operators to states.
It has two arguments and is used in the following combinations:
{\bf (i)} \f{LET OPAPPLY(}{\it operator, state}\f{) =} {\it state};
This is to define a elementary
action of an operator on a state in analogy to the way
elementary commutation relations are introduced to the system.
\begin{verbatim}
SCALOP A; STATE U;
FOR ALL N,P LET OPAPPLY((A(N),U(P))= EXP(I*N*P)*U(P);
\end{verbatim}
{\bf (ii)} \f{LET OPAPPLY(}{\it state, state}\f{) =} {\it scalar exp.};
This form is to define scalar products between states and normalisation
conditions.
\begin{verbatim}
STATE U;
FOR ALL N,M LET OPAPPLY(U(N),U(M)) = IF N=M THEN 1 ELSE 0;
\end{verbatim}
{\bf (iii)} {\it state} \f{:= OPAPPLY(}{\it operator expression, state});
In this way, the action of an operator expression on a given state
is calculated using elementary relations defined as explained in {\bf
(i)}. The result may be assigned to a different state vector.
{\bf (iv)} \f{OPAPPLY(}{\it state}\f{, OPAPPLY(}{\it operator expression,
state}\f{))}; This is the way how to calculate matrix elements of
operator
expressions. The system proceeds in the following way: first the
rightmost operator is applied on the right state, which means that the
system tries
to find an elementary relation which match the application of the
operator on the state. If it fails
the system tries to apply the leftmost operator of the expression on the
left state using the adjoint representations. If this fails also,
the system prints out a warning message and stops the evaluation.
Otherwise the next operator occuring in the expression is
taken and so on until the complete expression is applied. Then the
system
looks for a relation expressing the scalar product of the two
resulting states and prints out the final result. An example of such
a calculation is given in the test file.
The infix version of the \f{OPAPPLY} function is the vertical bar
$\mid$. It is \underline{right} associative and placed in the
precedence
list just above the minus ($-$) operator.