[Article] Transgressing the Boundaries: Toward a by Sokal A.D. PDF

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T h e Bose-Mesner algebra A acts on H from the left in such a way t h a t dry = {ex\aey} for a = (axy) and (av)x = jTj yex £ A and v = (vx) £ H- In particular, ACX = / yr^x ^ £y • axyvy, 48 Lemma 1 Let A be the Bose-Mesner algebra of a distance-regular graph X. Then, for any x G X we have (j>(a) = — Tr(a) = (ex\aex), a G A. (3) I I The proof is easy by (2). Obviously, defined in (3) becomes a state on A and we come to a quantum probability space (A,). In general, a probability measure // on R is called the distribution of a = a* G A under if (am)= J tmfi(dx), m = 0,1,2,- ••.

Bilinear form on W*t „ 2 x W U l ,„ 2 is denoted by ((•, -)}c. Then The oo (($,v))c = ^ I ! m ! -. e (^' + ™'):, mm . 5) Examples The combinations of two functions out of following examples are applicable to our setting. 1. Consider «(r) = «*(r) = er. Then it is obvious to check that consditions (UO) (U2) (U3) are satisfied. This example produces to the Hida-Kubo-Takenaka space over the real Gaussian space. See [14, 21, 22, 26]. 2. l . It is easy to check that u belongs to C+ii^ and satisfies conditions (UO) (U2) (U3).

L2) There exists a nonnegative constant C such that W{x,y)\2 < C M W i p J M M i p J for any (*,„) € £*puC x £^c. yJIuidili^J-iuddKli^,)-*. 1) for a function « € C+ i og . Define the space T>aiiU2 of test functions on £* x £* to be the projective limit of T>PltP2 as pi,p2 ->• °o. Let 27* „ be the dual space ofPUl,U2. Remark. This construction is motivated by Lee [25] and Asai et al. [5, 7, 8]. See also [15, 23] and references cited therein. Asai et al. [5, 7, 8] and Gannoun et al. [11] have considered the case of u2 = 1, independently.

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[Article] Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity by Sokal A.D.

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