Feynman Path Integral Formulation

7.2 Lattice Weak Field Expansion and Transverse-Traceless Modes 231counterparts h μν (x), which involves a sequence of non-trivial ω-dependent transformations,expressed by the matrices S and T . One more important aspect of theprocess is the disappearance of redundant lattice variables (five in the case of thehypercubic lattice), whose dynamics turns out to be trivial, in the sense that theassociated degrees of freedom are non-propagating.It is easy to see that the sequence of transformations expressed by the matricesS of Eq. (7.10) and T of Eq. (7.16), and therefore ultimately relating the latticefluctuations ε i (n) to their continuum counterparts h μν (x), just reproduces the expectedrelationship between lattice and continuum fields. On the one hand one hasg μν = η μν + h μν , where η μν is the flat metric. At the same time one has fromEq. (6.3) for each simplex within a given hypercubeg ij = 1 2 (l2 0i + l 2 0 j − l 2 ij) . (7.25)By inserting l i = l 0 i (1 + ε i ), with l 0 i = 1, √ 2, √ 3,2 for the body principal (i =1,2,4,8), face diagonal (i = 3,5,6,9,10,12), body diagonal (i = 7,11,13,14) andhyperbody diagonal (i = 15), respectively, one gets for example (1+ε 1 ) 2 = 1+h 11 ,(1 + ε 3 ) 2 = 1 + 1 2 (h 11 + h 22 )+h 12 etc., which in turn can then be solved for the ε’sin terms of the h μν ’s. One would then obtainε 1 = −1 +[1 +h 11 ] 1/2ε 3 = −1 +[1 + 1 2 (h 11 + h 22 )+h 12 )] 1/2ε 7 = −1 +[1 + 1 3 (h 11 + h 22 + h 33 )+ 2 3 (h 12 + h 23 + h 13 )] 1/2ε 15 = −1 +[1 + 1 4 (h 11 + h 22 + h 33 + h 44 )+ 3 4 (h 12 + h 13 + h 14 + h 23 + h 24 + h 34 )] 1/2 ,(7.26)and so on for the other edges, by suitably permuting indices. These relations canthen be expanded out for weak h, giving for exampleε 1 = 1 2 h 11 + O(h 2 )ε 3 = 1 2 h 12 + 1 4 (h 11 + h 22 )+O(h 2 )ε 7 = 1 6 (h 12 + h 13 + h 23 )+ 1 6 (h 23 + h 13 + h 12 )+ 1 6 (h 11 + h 22 + h 33 )+O(h 2 ) ,(7.27)and so on. The above correspondence between the ε’s and the h μν are the underlyingreason for the existence of the rotation matrices S and T of Eqs. (7.10) and (7.16),with one further important amendment: on the hypercubic lattice four edges within agiven simplex are assigned to one vertex, while the remaining six edges are assignedto neighboring vertices, and require therefore a translation back to the base vertex of