Connection, horizontal lift, and berry phase

In summary, connection in science refers to the mathematical framework used to describe how quantities change on curved surfaces or in higher-dimensional spaces. Horizontal lift is the process of preserving the horizontal direction of a curve or surface when lifted to another space. Berry phase is a quantum phenomenon that has important implications for quantum computing and topological phases of matter. Connection is closely related to parallel transport, which is the process of moving a vector or tensor while keeping it parallel to itself. The applications of horizontal lift and berry phase can be seen in fields such as physics, engineering, and quantum computing.
  • #1
wdlang
307
0
i am now reading the PRL paper by simon on berry phase

under adiabatic approximation, the wave function evolves as

\langle \psi | d/dt | \psi \rangle=0.

how to relate this equation to the connection on a fiber bundle?

how to understand that the wave function is parallel transported?
 
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  • #2
Your questions are answered in http://pra.aps.org/abstract/PRA/v43/i3/p1206_1" .
 
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  • #3
element4 said:
Your questions are answered in http://pra.aps.org/abstract/PRA/v43/i3/p1206_1" .

thanks a lot

i will have a look
 
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What is the concept of connection in science?

The concept of connection in science refers to the mathematical framework used to describe how quantities change as one moves along a curved surface or in a higher-dimensional space. It is also known as differential geometry and is essential in understanding the behavior of physical systems.

How is horizontal lift defined?

Horizontal lift is a geometric concept that refers to a process of lifting a curve or surface from one space to another in a way that preserves its horizontal direction. This is important in understanding the behavior of particles or objects moving along a curved path.

What is the significance of berry phase in quantum mechanics?

Berry phase, also known as geometric phase, is a phenomenon in quantum mechanics where a particle's wave function accumulates a phase shift when it is taken around a closed path in a parameter space. This phase is independent of the particle's energy and has important implications for quantum computing and topological phases of matter.

How is connection related to parallel transport?

Connection is closely related to parallel transport, which is the process of moving a vector or tensor along a curve while keeping it parallel to itself at each point. The connection determines how the vector or tensor changes as it is transported along the curve.

What are some real-world applications of horizontal lift and berry phase?

Horizontal lift and berry phase have a wide range of applications in various fields of science and technology. In physics, they are used to understand the behavior of particles in curved spacetime and to study topological phases of matter. In engineering, they are used in control theory and robotics. They also have applications in quantum computing, quantum information theory, and geometric mechanics.

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