AN UNCERTAINTY PRINCIPLE RELATED TO THE EUCLIDEAN MOTION GROUP

We show that a well-known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group. 1. Uncertainty principles related to Lie group representations Let G be a Lie group with Lie algebra g, and let (n, H) be a unitary representation of G. Then each element Xe g generates a closed, skew-adjoint operator n(X) on H by n(exptX)x-x n(X)x = hm '-o t with domain D(n(X)) consisting of all xe H for which the limit exists. The uncertainty principle related to n says that for operators generated by X, Y and [X, Y] the following holds ''n(X)x''''n(Y)x''>^'(n([X,Y])x,x)' for all xeD(n(X))nD(n(Y))nD(n([X, Y])). We would like to advocate this as a natural way to achieve uncertainty principles. It was first proposed in Kraus [4] for Lie groups with three dimensions or fewer, and the dimension constraint was recently removed by Christensen in [2]. For example, the classical Heisenberg uncertainty principle for functions on Un is easily derived in this way from the Schrodinger representation of the Heisenberg group (see [3, p. 212]). It is the purpose of this note to point out that the uncertainty principle for the circle, which was motivated by Breitenberger [1] and further discussed in [5; 6; 7; 8; 9; 10] is obtained similarly from the principal series representation of the Euclidean motion group of U2. * Corresponding author; e-mail: vepjan@math.ku.dk Mathematical Proceedings of the Royal Irish Academy, 104A (2), 249-252 (2004) © Royal Irish Academy This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms 250 Mathematical Proceedings of the Royal Irish Academy 2. The Euclidean motion group and a unitary representation Let G be the Euclidean motion group G=j(r,z)=(^or ^|rER,zecj Its Lie algebra is fl={(j j)|reR,zec} Let H be the Hubert space H = L2(J) of square integrable functions on the circle T = {s e C| 's' = 1}, with inner product (/", g) = JT f(t)g(t)dt. As in [1 1, chapter V] the following defines a unitary representation of G on ^i: 7Tfl(r, z)/(?) = ^Re(z^/(^-^), (j e T) where a e C. For simplicity we assume in the following that a = l, which is sufficient for our purpose. The representation n' will be denoted n. 3. Operators generated from the representation We now generate three operators from elements of the Lie algebra g. Let X,YU r2egbe *-G s), ".-(s i) ^(o ?) then exp(ijr)=^ J), exParl)=^ i) and QxV(tY2)=(^ ^ and we then get 7t(X)f(s) = lim i-+0 / i->0 ? where /'(a) = jtf{eus). This operator has domain {/eL2(T)|?h-»/V0 absolutely continuous with /7 e L2(T)} (3.1) Also rv^^ v hm 1*0, t)f(s) f(s) hm e^s)fis) fis) . ,_, . 7t( rv^^ Fj)/^) = v hm i-o t t-+o t and /VUM r hm 40, iY)/(j) /(j) hm r e^fis) fis) . . . _. _ . . ni /VUM Y2)fis) = hm r t^o i /->o t when s=e10 . Both of these operators are defined on H. This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms Christensen and Schlichtkrull Euclidean motion group 251 4. The uncertainty principle Since [X,YX] = Y21 [X,Y2]=-Yl the uncertainty principle gives 1(7*7, )/,/>| = '(n([X, Y2])fJ)' The first author would like to thank the people at NUI Galway for the help and friendliness he received while writing his thesis. References [1] E. Breitenberger, Uncertainty measures and uncertainty relations for angle observables, Foundations of Physics 15 (1985), 353-64. [2] J.G. Christensen, The uncertainty principle for operators determined by Lie groups, to appear in Journal of Fourier Analysis and Applications 10 (2004), 541-4. [3] G.B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, Journal of Fourier Analysis and Applications 3 (1997), 207-38. [4] K. Kraus, A further remark on uncertainty relations, Zeitschrift fur Physik 201 (1967), 134-41. [5] F.J. Narcowich and J.D. Ward, Wavelets associated with periodic basis functions, Applied and Computational Harmonic Analysis 3 (1996), 40-56. [6] J. Prestin and E. Quak, Optimal functions for a periodic uncertainty principle and multiresolution analysis, The Proceedings of the Edinburgh Mathematical Society 42 (1999), 225-42. [7] J. Prestin, E. Quak, H. Rauhut and K. Selig, On the connection of uncertainty principles for functions on the circle and on the real line, Journal of Fourier Analysis and Applications 9 (2003), 387-409. [8] H. Rauhut, An uncertainty principle for periodic functions, Diplomarbeit, Technische Universitat Munchen, 2001. Available online at: http://www.ma.tum.de/gkaam/personen/rauhut/ This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms 252 Mathematical Proceedings of the Royal Irish Academy [9] M. Rosier and M. Voit, An uncertainty principle for ultraspherical expansions, Journal of Mathematical Analysis and Applications 209 (1997), 624-34. [10] K.K. Selig, Uncertainty principles revisited, Electronic Transactions on Numerical Analysis 14 (2002), 164-76 (electronic). [1 1] M. Sugiura, Unitary representations and harmonic analysis, North-Holland Mathematical Library, 44, North-Holland Publishing Co., 1990. This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms