<s>
In	O
Hamiltonian	O
mechanics	O
,	O
the	O
linear	B-Algorithm
canonical	I-Algorithm
transformation	I-Algorithm
(	O
LCT	O
)	O
is	O
a	O
family	O
of	O
integral	B-Algorithm
transforms	I-Algorithm
that	O
generalizes	O
many	O
classical	O
transforms	O
.	O
</s>
<s>
The	O
LCT	O
generalizes	O
the	O
Fourier	B-Algorithm
,	O
fractional	O
Fourier	B-Algorithm
,	O
Laplace	O
,	O
Gauss	B-Algorithm
–	I-Algorithm
Weierstrass	I-Algorithm
,	O
Bargmann	O
and	O
the	O
Fresnel	O
transforms	O
as	O
particular	O
cases	O
.	O
</s>
<s>
The	O
name	O
"	O
linear	B-Algorithm
canonical	I-Algorithm
transformation	I-Algorithm
"	O
is	O
from	O
canonical	B-Algorithm
transformation	I-Algorithm
,	O
a	O
map	O
that	O
preserves	O
the	O
symplectic	O
structure	O
,	O
as	O
SL2(R )	O
can	O
also	O
be	O
interpreted	O
as	O
the	O
symplectic	O
group	O
Sp2	O
,	O
and	O
thus	O
LCTs	O
are	O
the	O
linear	O
maps	O
of	O
the	O
time	O
–	O
frequency	O
domain	O
which	O
preserve	O
the	O
symplectic	O
form	O
,	O
and	O
their	O
action	O
on	O
the	O
Hilbert	O
space	O
is	O
given	O
by	O
the	O
Metaplectic	O
group	O
.	O
</s>
<s>
The	O
basic	O
properties	O
of	O
the	O
transformations	O
mentioned	O
above	O
,	O
such	O
as	O
scaling	B-Algorithm
,	O
shift	O
,	O
coordinate	O
multiplication	O
are	O
considered	O
.	O
</s>
<s>
Any	O
linear	B-Algorithm
canonical	I-Algorithm
transformation	I-Algorithm
is	O
related	O
to	O
affine	O
transformations	O
in	O
phase	O
space	O
,	O
defined	O
by	O
time-frequency	O
or	O
position-momentum	O
coordinates	O
.	O
</s>
<s>
Many	O
classical	O
transforms	O
are	O
special	O
cases	O
of	O
the	O
linear	B-Algorithm
canonical	I-Algorithm
transform	I-Algorithm
:	O
</s>
<s>
Scaling	B-Algorithm
,	O
,	O
corresponds	O
to	O
scaling	B-Algorithm
the	O
time	O
and	O
frequency	O
dimensions	O
inversely	O
(	O
as	O
time	O
goes	O
faster	O
,	O
frequencies	O
are	O
higher	O
and	O
the	O
time	O
dimension	O
shrinks	O
)	O
:	O
</s>
<s>
Composition	O
of	O
LCTs	O
corresponds	O
to	O
multiplication	O
of	O
the	O
corresponding	O
matrices	O
;	O
this	O
is	O
also	O
known	O
as	O
the	O
additivity	O
property	O
of	O
the	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
(	O
WDF	O
)	O
.	O
</s>
<s>
Paraxial	O
optical	O
systems	O
implemented	O
entirely	O
with	O
thin	O
lenses	O
and	O
propagation	O
through	O
free	O
space	O
and/or	O
graded-index	O
(	O
GRIN	O
)	O
media	O
,	O
are	O
quadratic-phase	O
systems	O
(	O
QPS	O
)	O
;	O
these	O
were	O
known	O
before	O
Moshinsky	O
and	O
Quesne	O
(	O
1974	O
)	O
called	O
attention	O
to	O
their	O
significance	O
in	O
connection	O
with	O
canonical	B-Algorithm
transformations	I-Algorithm
in	O
quantum	O
mechanics	O
.	O
</s>
<s>
The	O
effect	O
of	O
any	O
arbitrary	O
QPS	O
on	O
an	O
input	O
wavefield	O
can	O
be	O
described	O
using	O
the	O
linear	B-Algorithm
canonical	I-Algorithm
transform	I-Algorithm
,	O
a	O
particular	O
case	O
of	O
which	O
was	O
developed	O
by	O
Segal	O
(	O
1963	O
)	O
and	O
Bargmann	O
(	O
1961	O
)	O
in	O
order	O
to	O
formalize	O
Fock	O
's	O
(	O
1928	O
)	O
boson	O
calculus	O
.	O
</s>
<s>
In	O
quantum	O
mechanics	O
,	O
linear	B-Algorithm
canonical	I-Algorithm
transformations	I-Algorithm
can	O
be	O
identified	O
with	O
the	O
linear	O
transformations	O
which	O
mix	O
the	O
momentum	O
operator	O
with	O
the	O
position	O
operator	O
and	O
leave	O
invariant	O
the	O
canonical	O
commutation	O
relations	O
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
it	O
may	O
be	O
possible	O
to	O
establish	O
a	O
relation	O
between	O
some	O
properties	O
of	O
the	O
elementary	O
fermion	O
in	O
the	O
Standard	B-General_Concept
Model	I-General_Concept
of	O
particle	O
physics	O
and	O
spin	O
representation	O
of	O
linear	B-Algorithm
canonical	I-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
In	O
this	O
approach	O
,	O
the	O
electric	O
charge	O
,	O
weak	O
hypercharge	O
and	O
weak	O
isospin	O
of	O
the	O
particles	O
are	O
expressed	O
as	O
linear	O
combinations	O
of	O
some	O
operators	O
defined	O
from	O
the	O
generators	O
of	O
the	O
Clifford	O
algebra	O
associated	O
with	O
the	O
spin	O
representation	O
of	O
linear	B-Algorithm
canonical	I-Algorithm
transformations	I-Algorithm
.	O
</s>
