Infinitesimal Rotations and Constraints in Physics

Eberhard Zeidler
Additional contact information
Eberhard Zeidler: Max Planck Institute for Mathematics in the Sciences

Chapter 6 in Quantum Field Theory III: Gauge Theory, 2011, pp 371-423 from Springer

Abstract: Abstract The operator A:E 3→E 3 is called unitary iff it is linear and it respects the inner product, that is, $$\langle A\mathbf{x}|A\mathbf{y}\rangle = \langle \mathbf{x}|\mathbf{y}\rangle \qquad \mbox{for all}\quad \mathbf{x}, \mathbf{y}\in E_3.$$ The symbol U(E 3) denotes the set of all unitary operators A:E 3→E 3. We have $$A \in U(E_3)\qquad \mbox{iff}\qquad A^\dagger A =I.$$ In fact, it follows from (6.1) that $$\langle \mathbf{x}|A^\dagger A\mathbf{y}\rangle = \langle A\mathbf{x}|A\mathbf{y}\rangle =\langle \mathbf{x}|\mathbf{y}\rangle \qquad \mbox{for all}\quad \mathbf{x}, \mathbf{y}\in E_3.$$ Hence A † A=I. Conversely, A † A=I implies (6.1). If A∈U(E 3), then det (A)=±1. In fact, I=A † A implies 1=det I=det A †det A=(det A)†det A=|det A|2.

Keywords: Gauge Theory; Rigid Body; Lagrange Equation; Celestial Body; Nonholonomic Constraint (search for similar items in EconPapers)
Date: 2011
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-22421-8_7

Ordering information: This item can be ordered from
http://www.springer.com/9783642224218

DOI: 10.1007/978-3-642-22421-8_7

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().