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Up: Results for the Galilean
Previous: Second order
Fourth order
At fourth order, imposing either Galilean or gauge symmetry forces 12 dependent coupling
constants to be specific linear combinations of 3 independent ones:
![$\displaystyle \bm{C_{00,4000}^{0000}}$](img893.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{3 }{2 \sqrt{5}}\bm{C_{00,2202}^{2202}} ,$](img894.png) |
(126) |
![$\displaystyle \bm{C_{00,2000}^{2000}}$](img895.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{1}{2} \sqrt{5} \bm{C_{00,2202}^{2202}} ,$](img896.png) |
(127) |
![$\displaystyle {}{C_{00,3101}^{1101}}$](img897.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\tfrac{6 }{\sqrt{5}} \bm{C_{00,2202}^{2202}} ,$](img898.png) |
(128) |
![$\displaystyle \bm{C_{00,3110}^{1110}}$](img899.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -2 \sqrt{\tfrac{3}{5}} {}{C_{00,2212}^{2212}}-\tfrac{7 }{\sqrt{5}}{}{C_{00,4211}^{0011}} ,$](img900.png) |
(129) |
![$\displaystyle \bm{C_{00,3111}^{1111}}$](img901.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\tfrac{6 }{\sqrt{5}}{}{C_{00,2212}^{2212}} ,$](img902.png) |
(130) |
![$\displaystyle \bm{C_{00,3112}^{1112}}$](img903.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -2 \sqrt{3} {}{C_{00,2212}^{2212}}-\tfrac{14 }{5}{}{C_{00,4211}^{0011}} ,$](img904.png) |
(131) |
![$\displaystyle \bm{C_{00,3312}^{1112}}$](img905.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -2 \sqrt{\tfrac{7}{15}} {}{C_{00,4211}^{0011}} ,$](img906.png) |
(132) |
![$\displaystyle {}{C_{00,4011}^{0011}}$](img907.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{3}{2} \sqrt{\tfrac{3}{5}} {}{C_{00,2212}^{2212}}+\tfrac{7 }{4 \sqrt{5}}{}{C_{00,4211}^{0011}} ,$](img908.png) |
(133) |
![$\displaystyle {}{C_{00,2011}^{2011}}$](img909.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{1}{2} \sqrt{15} {}{C_{00,2212}^{2212}}+\tfrac{7}{12} \sqrt{5} {}{C_{00,4211}^{0011}} ,$](img910.png) |
(134) |
![$\displaystyle {}{C_{00,2211}^{2011}}$](img911.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{7 }{3}{}{C_{00,4211}^{0011}} ,$](img912.png) |
(135) |
![$\displaystyle {}{C_{00,2211}^{2211}}$](img913.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \sqrt{\tfrac{3}{5}} {}{C_{00,2212}^{2212}}+\tfrac{7 }{3 \sqrt{5}}{}{C_{00,4211}^{0011}} ,$](img914.png) |
(136) |
![$\displaystyle {}{C_{00,2213}^{2213}}$](img915.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \sqrt{\tfrac{7}{5}} {}{C_{00,2212}^{2212}}+\tfrac{1}{2} \sqrt{\tfrac{21}{5}} {}{C_{00,4211}^{0011}} .$](img916.png) |
(137) |
At this order, there are 3 stand-alone Galilean and gauge invariant terms:
and 3 Galilean and gauge invariant linear combinations of terms, corresponding to
the 3 independent coupling constants:
Altogether, these 6 free coupling constants (3 unrestricted and 3 independent)
occur in both the Galilean and gauge invariant EDF at fourth order,
cf. Table 6.
Apart from these 6 free and 12 dependent coupling constants,
the gauge invariance requires that all the remaining 27
coupling constants are equal to zero. These 27 constants are allowed to
be non-zero if the Galilean symmetry is imposed instead of the
full gauge invariance. Then, there are 18 dependent coupling
constants that are forced to be linear combinations of 9 independent
ones:
![$\displaystyle \bm{C_{20,2000}^{0000}}$](img931.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -{}{C_{20,1101}^{1101}} ,$](img932.png) |
(144) |
![$\displaystyle \bm{C_{22,2202}^{0000}}$](img933.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -{}{C_{22,1101}^{1101}} ,$](img934.png) |
(145) |
![$\displaystyle \bm{C_{20,1110}^{1110}}$](img935.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\tfrac{1}{3}{}{C_{20,2011}^{0011}}-\tfrac{1}{3} \sqrt{5} {}{C_{20,2211}^{0011}} ,$](img936.png) |
(146) |
![$\displaystyle \bm{C_{22,1112}^{1110}}$](img937.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\bm{C_{22,1112}^{1111}}-\tfrac{2 }{\sqrt{7}}\bm{C_{22,1112}^{1112}}$](img938.png) |
|
|
|
![$\displaystyle -2 \sqrt{\tfrac{15}{7}} {}{C_{22,2213}^{0011}} ,$](img939.png) |
(147) |
![$\displaystyle \bm{C_{20,1111}^{1111}}$](img940.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{1}{2} \sqrt{\tfrac{5}{3}} {}{C_{20,2211}^{0011}}-\tfrac{1}{\sqrt{3}}{}{C_{20,2011}^{0011}} ,$](img941.png) |
(148) |
![$\displaystyle \bm{C_{22,1111}^{1111}}$](img942.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\tfrac{1}{\sqrt{3}}\bm{C_{22,1112}^{1111}}-3 \sqrt{\tfrac{3}{7}} \bm{C_{22,1112}^{1112}}$](img943.png) |
|
|
|
![$\displaystyle -2 \sqrt{\tfrac{5}{7}} {}{C_{22,2213}^{0011}} ,$](img944.png) |
(149) |
![$\displaystyle \bm{C_{20,1112}^{1112}}$](img945.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\tfrac{1}{3} \sqrt{5} {}{C_{20,2011}^{0011}}-\tfrac{1}{6}{}{C_{20,2211}^{0011}} ,$](img946.png) |
(150) |
![$\displaystyle {}{C_{22,2011}^{0011}}$](img947.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{2 }{3}\bm{C_{22,1112}^{1111}}-\tfrac{2 }{\sqrt{7}}\bm{C_{22,1112}^{1112}}$](img948.png) |
|
|
|
![$\displaystyle +\sqrt{\tfrac{5}{21}} {}{C_{22,2213}^{0011}} ,$](img949.png) |
(151) |
![$\displaystyle {}{C_{22,2211}^{0011}}$](img950.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{1}{6} \sqrt{5} \bm{C_{22,1112}^{1111}}+\sqrt{\tfrac{5}{7}} \bm{C_{22,1112}^{1112}}$](img951.png) |
|
|
|
![$\displaystyle +\tfrac{8 }{\sqrt{21}}{}{C_{22,2213}^{0011}} ,$](img952.png) |
(152) |
![$\displaystyle {}{C_{22,2212}^{0011}}$](img953.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{1}{2} \sqrt{3} \bm{C_{22,1112}^{1111}}+3 \sqrt{\tfrac{3}{7}} \bm{C_{22,1112}^{1112}}$](img954.png) |
|
|
|
![$\displaystyle +2 \sqrt{\tfrac{5}{7}} {}{C_{22,2213}^{0011}} ,$](img955.png) |
(153) |
![$\displaystyle \bm{C_{31,1111}^{0000}}$](img956.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle {}{C_{31,0011}^{1101}} ,$](img957.png) |
(154) |
![$\displaystyle \bm{C_{11,3111}^{0000}}$](img958.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\sqrt{\tfrac{3}{5}} {}{C_{11,2212}^{1101}} ,$](img959.png) |
(155) |
![$\displaystyle \bm{C_{11,1111}^{2000}}$](img960.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\sqrt{\tfrac{5}{3}} {}{C_{11,2212}^{1101}} ,$](img961.png) |
(156) |
![$\displaystyle \bm{C_{11,1111}^{2202}}$](img962.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{1}{\sqrt{3}}{}{C_{11,2212}^{1101}} ,$](img963.png) |
(157) |
![$\displaystyle \bm{C_{11,1112}^{2202}}$](img964.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle {}{C_{11,2212}^{1101}} ,$](img965.png) |
(158) |
![$\displaystyle {}{C_{11,2011}^{1101}}$](img966.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\sqrt{\tfrac{5}{3}} {}{C_{11,2212}^{1101}} ,$](img961.png) |
(159) |
![$\displaystyle {}{C_{11,2211}^{1101}}$](img967.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle \tfrac{1}{\sqrt{3}}{}{C_{11,2212}^{1101}} ,$](img963.png) |
(160) |
![$\displaystyle {}{C_{11,0011}^{3101}}$](img968.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\sqrt{\tfrac{3}{5}} {}{C_{11,2212}^{1101}} .$](img969.png) |
(161) |
Finally,
we list 9 combinations of terms that are
invariant with respect to the Galilean symmetry and correspond to
the independent coupling constants:
![$\displaystyle {}{G_{22,1112}^{1111}}$](img970.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\tfrac{1}{\sqrt{3}}\bm{T_{22,1111}^{1111}}-\bm{T_{22,1112}^{1110...
... \sqrt{5} {}{T_{22,2211}^{0011}}+\tfrac{1}{2} \sqrt{3} {}{T_{22,2212}^{0011}} ,$](img971.png) |
(162) |
![$\displaystyle {}{G_{22,1112}^{1112}}$](img972.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -3 \sqrt{\tfrac{3}{7}} \bm{T_{22,1111}^{1111}}-\tfrac{2 }{\sqrt{7...
...112}^{1110}}+\bm{T_{22,1112}^{1112}}-\tfrac{2 }{\sqrt{7}}{}{T_{22,2011}^{0011}}$](img973.png) |
|
|
|
![$\displaystyle +\sqrt{\tfrac{5}{7}} {}{T_{22,2211}^{0011}}+3 \sqrt{\tfrac{3}{7}} {}{T_{22,2212}^{0011}} ,$](img974.png) |
(163) |
![$\displaystyle {}{G_{20,1101}^{1101}}$](img975.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle {}{T_{20,1101}^{1101}}-\bm{T_{20,2000}^{0000}} ,$](img976.png) |
(164) |
![$\displaystyle {}{G_{22,1101}^{1101}}$](img977.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle {}{T_{22,1101}^{1101}}-\bm{T_{22,2202}^{0000}} ,$](img978.png) |
(165) |
![$\displaystyle {}{G_{31,0011}^{1101}}$](img979.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle {}{T_{31,0011}^{1101}}+\bm{T_{31,1111}^{0000}} ,$](img980.png) |
(166) |
![$\displaystyle {}{G_{11,2212}^{1101}}$](img981.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\sqrt{\tfrac{5}{3}} \bm{T_{11,1111}^{2000}}+\tfrac{1}{\sqrt{3}}\...
...{\tfrac{5}{3}} {}{T_{11,2011}^{1101}}+\tfrac{1}{\sqrt{3}}{}{T_{11,2211}^{1101}}$](img982.png) |
|
|
|
![$\displaystyle -\sqrt{\tfrac{3}{5}} \bm{T_{11,3111}^{0000}}+{}{T_{11,2212}^{1101}}-\sqrt{\tfrac{3}{5}} {}{T_{11,0011}^{3101}} ,$](img983.png) |
(167) |
![$\displaystyle {}{G_{20,2011}^{0011}}$](img984.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\tfrac{1}{3}\bm{T_{20,1110}^{1110}}-\tfrac{1}{\sqrt{3}}\bm{T_{20...
...^{1111}}-\tfrac{1}{3} \sqrt{5} \bm{T_{20,1112}^{1112}}+{}{T_{20,2011}^{0011}} ,$](img985.png) |
(168) |
![$\displaystyle {}{G_{20,2211}^{0011}}$](img986.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -\tfrac{1}{3} \sqrt{5} \bm{T_{20,1110}^{1110}}+\tfrac{1}{2} \sqrt...
..._{20,1111}^{1111}}-\tfrac{1}{6}\bm{T_{20,1112}^{1112}}+{}{T_{20,2211}^{0011}} ,$](img987.png) |
(169) |
![$\displaystyle {}{G_{22,2213}^{0011}}$](img988.png) |
![$\displaystyle =$](img12.png) |
![$\displaystyle -2 \sqrt{\tfrac{5}{7}} \bm{T_{22,1111}^{1111}}-2 \sqrt{\tfrac{15}...
...frac{5}{21}} {}{T_{22,2011}^{0011}}+\tfrac{8 }{\sqrt{21}}{}{T_{22,2211}^{0011}}$](img989.png) |
|
|
|
![$\displaystyle +2 \sqrt{\tfrac{5}{7}} {}{T_{22,2212}^{0011}}+{}{T_{22,2213}^{0011}} .$](img990.png) |
(170) |
Next: Sixth order
Up: Results for the Galilean
Previous: Second order
Jacek Dobaczewski
2008-10-06