A straight string of tension and length has one end held at a fixed point by a…

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Beyond the Standard Model · Beyond the Standard Model

Symmetry and Flavour, Ch The relativistic string: action, light-cone quantisation, spectrum and D-branes

Problem

A straight string of tension

T_0

and length

\ell

has one end held at a fixed point by a D0-brane and the other end free. It rotates rigidly in a plane about the held end. Let

r

be the distance of a point of the string from the held end. ((a)) Find the speed

v(r)

of the point at

r

. ((b)) Each element

\mathrm{d}r

carries energy

T_0\,\mathrm{d}r/\sqrt{1-v^2}

and momentum

T_0v\,\mathrm{d}r/\sqrt{1-v^2}

, as in the rotating string with two free ends. Find the energy

E

and the angular momentum

J

in terms of

T_0

and

\ell

. ((c)) Show that

J = 2\alpha'E^2

, with

\alpha' = 1/2\pi T_0

, and explain in one sentence why the slope is twice that of a string with two free ends. ((d)) For

T_0=0.20\,\mathrm{GeV^2}

and

E=1.5\,\mathrm{GeV}

, find

\ell

\mathrm{fm}

, using

\hbar c=0.1973\,\mathrm{GeV\,fm}

, and

J

(a pure number, since

\hbar=1

). Compare

J

with that of a string with two free ends and the same energy.