Physics LettersA 164 (1992) 133—137 PHYSICS LETTERS A North-Holland A group theoretic treatment of the geometric phase

We define a new unitary operator in the Hubert space of a quantum system which parallel transports the state of the system along an arbitrary curve in the projective Hubert space. This operator is geometrical even for an open curve in the sense that it depends uniquely only on the curve and is independent of the Hamiltonian. Using this, when the curve is closed, the geometric phases discovered by Pancharatnam, Berry and Aharonov—Anandanare obtained. The geometric phase was first discovered by Pan- space ~, which is the set of rays of the Hubert space charatnam [1] for the cyclic evolution of a photon W, with respect to a connection arising from the in-polarization state due to a sequence offiltering mea- ner product in ~*‘. If * = C”, then ~ is CP”~,e.g. surements. Berry [2] showed that during an adi- for n=2, ~=CP ’ is the two-dimensional sphere S2. abatic cyclic evolution of an energy eigenstateof any The Berry phase may be seen tobe the adiabatic limit quantum system there is a phase factor which is in- of this more general geometric phase, which has also dependent of the rate of evolution. This phase can been measured experimentally [61. be measured experimentally. The geometric mean- In this note we obtain the geometric phase using ing ofthis adiabatic geometricphase was pointed out an operator which has a geometric meaning even for by Simon [3] as parallel transport in a line bundle a non-cyclic evolution. This result is useful also be-corresponding to the eigenstateof a particular energy cause it is of group theoretic natureand does not ex