FeynCalc manual (development version)

PauliSimplify

PauliSimplify[exp] simplifies products of Pauli matrices and expands non-commutative products. Double indices and vectors are contracted. The order of the Pauli matrices is not changed.

See also

Overview, PauliSigma, PauliTrick.

Examples

CSIS[p1] . CSI[i] . CSIS[p2] 
 
PauliSimplify[%]

(σp1).σi.(σp2)\left(\overline{\sigma }\cdot \overline{\text{p1}}\right).\overline{\sigma }^i.\left(\overline{\sigma }\cdot \overline{\text{p2}}\right)

(σp1).σi.(σp2)\left(\overline{\sigma }\cdot \overline{\text{p1}}\right).\overline{\sigma }^i.\left(\overline{\sigma }\cdot \overline{\text{p2}}\right)

CSIS[p] . CSI[i, j, k] . CSIS[p] 
 
PauliSimplify[%]

(σp).σi.σj.σk.(σp)\left(\overline{\sigma }\cdot \overline{p}\right).\overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^k.\left(\overline{\sigma }\cdot \overline{p}\right)

p2σi.σj.σk+2pkσi.σj.(σp)2pjσi.σk.(σp)+2piσj.σk.(σp)-\overline{p}^2 \overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^k+2 \overline{p}^k \overline{\sigma }^i.\overline{\sigma }^j.\left(\overline{\sigma }\cdot \overline{p}\right)-2 \overline{p}^j \overline{\sigma }^i.\overline{\sigma }^k.\left(\overline{\sigma }\cdot \overline{p}\right)+2 \overline{p}^i \overline{\sigma }^j.\overline{\sigma }^k.\left(\overline{\sigma }\cdot \overline{p}\right)

PauliSimplify[CSIS[p] . CSI[i, j, k] . CSIS[p], PauliReduce -> False]

p2σi.σj.σk+2pkσi.σj.(σp)2pjσi.σk.(σp)+2piσj.σk.(σp)-\overline{p}^2 \overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^k+2 \overline{p}^k \overline{\sigma }^i.\overline{\sigma }^j.\left(\overline{\sigma }\cdot \overline{p}\right)-2 \overline{p}^j \overline{\sigma }^i.\overline{\sigma }^k.\left(\overline{\sigma }\cdot \overline{p}\right)+2 \overline{p}^i \overline{\sigma }^j.\overline{\sigma }^k.\left(\overline{\sigma }\cdot \overline{p}\right)

CSID[i, j, i] 
 
PauliSimplify[%]

σi.σj.σi\sigma ^i.\sigma ^j.\sigma ^i

3σjDσj3 \sigma ^j-D \sigma ^j

CSID[i, j, k, l, m, i] 
 
PauliSimplify[%]

σi.σj.σk.σl.σm.σi\sigma ^i.\sigma ^j.\sigma ^k.\sigma ^l.\sigma ^m.\sigma ^i

Dσj.σk.σl.σm3σj.σk.σl.σm+2σj.σk.σm.σl2σj.σl.σm.σk+2σk.σl.σm.σjD \sigma ^j.\sigma ^k.\sigma ^l.\sigma ^m-3 \sigma ^j.\sigma ^k.\sigma ^l.\sigma ^m+2 \sigma ^j.\sigma ^k.\sigma ^m.\sigma ^l-2 \sigma ^j.\sigma ^l.\sigma ^m.\sigma ^k+2 \sigma ^k.\sigma ^l.\sigma ^m.\sigma ^j