What kind of cases are geometric phases real? In what case is the geometric phase complex? If the geometric phase is complex, how do we make sense of it?
Real Geometric Phase:
When the system is Hamiltonian, (energy operator) is real and the quantum state's (or wave function's) development is entirely unitary, the geometric phase is usually real. Said another way, the geometric phase obtained upon finishing a closed path will be real if the evolution of the quantum system is represented by a real-valued phase factor (such as in the case of a spin-1/2 particle in a magnetic field with a real vector potential). Since real geometric phases may be directly correlated with measurable physical quantities, they are easy to comprehend.
Complex Geometric Phase:
In some cases, such as when non-trivial dynamics are involved in the quantum state's evolution and the phase factor has imaginary components, the geometric phase can become complex. This usually happens in systems where the Hamiltonian comprises complicated components rather than being completely real. For example, in the setting of electromagnetic, when a complex vector potential is present. Complex geometric phases suggest that a real-valued phase factor is insufficient to completely represent the quantum state's evolution. As an alternative, the phase factor can have imaginary components, which would result in a geometric phase with complex values.
Said in different way, the final phase = initial phase,
Because psi initial=psi final. (If you go around in closed path)
Then this comon phase can be used to divide the phase everywhere,
it ends up real at start and end points.
A magnetic field will always force some
Nonzero phase.
`Complex´ is a definition in math.
It depends on the not solved task to do the square-root of minus one.
[√(-1) = i only is equivalence / short form for the same, no calculation]
If the root is for to look for the factors which would produce the radicand, there are two solutions if on -1:
(+1)(-1) = -1 as well as (-1)(+1) = -1
Both left sided are the solutions if on not reformed math.
There a function in minimum takes place too!
[see the complete published article:`European Scientific Journal, vol.9, No. 33, november 2013, page 179 ... 183.]
But reformed math would not be ambiguous:
(+1)(+1) = ++1
(+1)(-1) = +-1
(-1)(+1) = -+1
(-1)(-1) = --1
Now the roots are to take as the left sided expressions.
A root of the product of two factors has to depend on two solutions!
Squaring written by exponent is the short form of multiplication by two factors. One factor cannot act by combination (multiplication) as two would. The second misses!
Please do not take a tool in math for to have solutions in other sciences if this tool got (accepted) taken out.