Using j h x a π o ? ψ π h ? j ψ a h x j h π x ψ

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Using J h X A ˜ π o ( * η ) | ψ = - ˜ π ( h ( η ))( J | ψ A h ( X ) , J h ˜ π ( X ) | ψ A η = - h ( η ) A ˜ π o ( h ( X )) | ψ , it is seen that J D = D J implies J h D = D J h . (B.14) Just as H is a Hilbert A -bimodule, H Cov is a Hilbert B -bimodule with the action of A , B ∈ B given by : X A | ψ A η ( A X ) A | ψ A ( η B ) . The spectral data {A , H , D ,χ, J } , also called (non-commutative) geome- try in [21], is said to be Morita equivalent to the geometry defined by {B , H Cov , D , Γ , J h } . Not only H Cov = M ⊗ A H ⊗ A M * , but also H = M * B H Cov B M ! 54
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References [1] Connes A.,”Noncommutative Geometry”, Acad.Press,London,1994. [2] Connes A., J.Math.Phys, 36 ,6194,1995. [3] Connes A., Comm.Math.Phys., 182 ,155,1996. [4] Connes A. and Lott J., Nucl.Phys.(Proc.Suppl.), 18B ,29,1990. [5] Fr¨ ohlich J. and Osterwalder K., Helv.Phys.Acta, 47 ,781,1974. [6] Coquereaux R., J.Geom.Phys., 6 ,425,1989. [7] Dubois-Violette M. and Masson T., Lett.Math.Phys., 37 ,467,1996. [8] Jayewardena C., Helv.Phys.Acta, 61 ,636,1988. [9] Kastler D. and Sch¨ucker T., Rev.Math.Phys., 8 ,205,1996. [10] Krajewski T., J.Geom.Phys., 28 ,2,1998. [11] Lizzi F.,Mangano G.,Miele G. and Sparano G., hep-th/9610035. [12] Mart´ ın C.P., Gracia-Bond´ ıa and V´ arilly J.,Phys.Rep. 294 ,363,1998. [13] Metha M.R., Phys.Rev.Lett., 65 ,1983,1990. Mod.Phys.Lett.A, 6 ,2811,1991. [14] Mignaco J.A.,Sigaud C.,Vanhecke F.J. and Da Silva A.R., Rev.Math.Phys., 9 ,689,1997. [15] Newman E. and Penrose R., J.Math.Phys., 7 ,863,1966. [16] Osterwalder K. and Schrader R., Helv.Phys.Acta, 46 ,277,1973. [17] Paschke M. and Sitarz A.,MZ-TH/96-40, Mainz,1996. [18] Staruszkiewicz A., J.Math.Phys., 8 ,2221,1967. [19] Vanhecke F.J.,math-phys/9903029 55
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[20] van Nieuwenhuizen P. and Waldron A., Phys.Lett.B, 389 ,29,1990 also hep-th/9610035 and Waldron A. hep-th/9702057. [21] V´ arilly J.C., ”An Introduction to Noncommutative Geometry”, Monsaraz lectures, Lisbon,1997. [22] V´ arilly J.C. and Gracia-Bond´ ıa J.M., J.Geom.and Phys., 12 ,223,1993. [23] Wu T.T. and Yang C.N., Phys.Rev., D12 ,3845,1975; Nucl.Phys, B107 ,365,1976. 56
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