Feynman Path Integral Formulation

7.2 Lattice Weak Field Expansion and Transverse-Traceless Modes 231
counterparts 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 the
process is the disappearance of redundant lattice variables (five in the case of the
hypercubic lattice), whose dynamics turns out to be trivial, in the sense that the
associated degrees of freedom are non-propagating.
It is easy to see that the sequence of transformations expressed by the matrices
S of Eq. (7.10) and T of Eq. (7.16), and therefore ultimately relating the lattice
fluctuations ε i (n) to their continuum counterparts h μν (x), just reproduces the expected
relationship between lattice and continuum fields. On the one hand one has
g μν = η μν + h μν , where η μν is the flat metric. At the same time one has from
Eq. (6.3) for each simplex within a given hypercube
g 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) and
hyperbody 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’s
in terms of the h μν ’s. One would then obtain
ε 1 = −1 +[1 +h 11 ] 1/2
ε 3 = −1 +[1 + 1 2 (h 11 + h 22 )+h 12 )] 1/2
ε 7 = −1 +[1 + 1 3 (h 11 + h 22 + h 33 )
+ 2 3 (h 12 + h 23 + h 13 )] 1/2
ε 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 can
then be expanded out for weak h, giving for example
ε 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 underlying
reason 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 a
given simplex are assigned to one vertex, while the remaining six edges are assigned
to neighboring vertices, and require therefore a translation back to the base vertex of