# 相对论质心

## 导论

${\displaystyle {\vec {x}}_{(nr)}(t)={\frac {\sum _{i=1}^{N}\,m_{i}\,{\vec {x}}_{i}(t)}{\sum _{i=1}^{N}\,m_{i}}}}$

i) 与总动量正则对易关系
ii) 在旋转中以三维向量变换，及
iii) 其位置有关系统粒子在空间中质量的分布。

1. Newton–Wigner–Pryce自旋中心或正则质心（Newton–Wigner量子位置算符的古典对应）。这是个三维向量${\displaystyle {\vec {\tilde {x}}}}$，符合有关牛顿质心的同样正则条件，即相空间中的帕松括号${\displaystyle \{{\tilde {x}}^{i},{\tilde {x}}^{j}\}=0}$。然而，没有四维向量${\displaystyle {\tilde {x}}^{\mu }=({\tilde {x}}^{o},{\vec {\tilde {x}}})}$能包括该三维向量作为空间部分，因此它并不确定定义世界线，只有依赖惯性参照系的伪世界线。
2. Fokker–Pryce惯性中心${\displaystyle {\vec {Y}}}$。这是四维向量${\displaystyle Y^{\mu }=(Y^{0},{\vec {Y}})}$的空间部分，因此能定义世界线，但非正则，即${\displaystyle \{Y^{i},Y^{j}\}\not =0}$
3. Møller能量中心${\displaystyle {\vec {R}}}$。定义是将牛顿质心中的静止质量${\displaystyle m_{i}}$改为它们的相对论能量。这亦非正则，即${\displaystyle \{R^{i},R^{j}\}\not =0}$，又不是四维向量的空间部分，即只能定义依赖参照系的伪世界线。

## 参见

1. ^ M.Pauri and G.M.Prosperi, Canonical Realizations of the Poincaré Group. I. General Theory, J.Math.Phys. ={16}, 1503 (1975). M.Pauri, Canonical (Possibly Lagrangian) Realizations of the Poincaré Group with Increasing Mass-Spin Trajectories, talk at the International Colloquium "Group Theoretical Methods in Physica", Cocoyoc, Mexico, 1980, edited by K.B.Wolf (Springer, Berlin, 1980)
2. ^ T.D.Newton and E.P.Wigner, Localized States for Elementary Systems, Rev.Mod.Phys. Vol 21, 400 1969.
3. ^ R.H.L.Pryce, The Mass-Centre in the Restricted Theory of Relativity and Its Connexion with the Quantum Theory of Elementary Particles, Proc.R.Soc.London, Ser A Vol 195, 62 (1948).
4. ^ A.D.Fokker, Relativiteitstheorie (Noordhoff, Groningen, 1929) p.171.
5. ^ C. Møller, Sur la dynamique des systemes ayant un moment angulaire interne, Ann.Inst.H.Poincaré vol {11}, 251 (1969); The Theory of Relativity (Oxford: Oxford University Press, 1957)
6. ^ G.N.Fleming, Covariant Position Operators, Spin and Locality, Phys.Rev. vol 137B, 188 (1965)
7. ^ A.J.Kalnay, The Localization Problem, in Studies in the Foundations, Methodology and Philosophy of Science, edited by M.Bunge (Springer, Berlin, 1971), vol.4
8. ^ M.Lorente and P.Roman, {General expressions for the position and spin operators of relativistic systems, J.Math.Phys. vol 15, 70 (1974).
9. ^ H.Sazdjian, {Position Variables in Classical Relativistic Hamiltonian Mechanics}, Nucl.Phys. vol B161,469 (1979).