Skip to main content
SpringerLink
Log in

Geometry of Schrödinger space-times II: particle and field probes of the causal structure

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We continue our study of the global properties of the z = 2 Schrödinger space-time. In particular, we provide a codimension 2 isometric embedding which naturally gives rise to the previously introduced global coordinates. Furthermore, we study the causal structure by probing the space-time with point particles as well as with scalar fields. We show that, even though there is no global time function in the technical sense (Schrödinger space-time being non-distinguishing), the time coordinate of the global Schrödinger coordinate system is, in a precise way, the closest one can get to having such a time function. In spite of this and the corresponding strongly Galilean and almost pathological causal structure of this space-time, it is nevertheless possible to define a Hilbert space of normalisable scalar modes with a well-defined time-evolution. We also discuss how the Galilean causal structure is reflected and encoded in the scalar Wightman functions and the bulk-to-bulk propagator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  2. K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  3. C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  4. J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  5. A. Adams, K. Balasubramanian and J. McGreevy, Hot spacetimes for cold atoms, JHEP 11 (2008) 059 [arXiv:0807.1111] [SPIRES].

    Article  ADS  Google Scholar 

  6. S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  7. S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  8. J. McGreevy, Holographic duality with a view toward many-body physics, arXiv:0909.0518 [SPIRES].

  9. M. Taylor, Non-relativistic holography, arXiv:0812.0530 [SPIRES].

  10. S.F. Ross and O. Saremi, Holographic stress tensor for non-relativistic theories, JHEP 09 (2009) 009 [arXiv:0907.1846] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  11. M. Blau, J. Hartong and B. Rollier, Geometry of Schrödinger space-times, global coordinates and harmonic trapping, JHEP 07 (2009) 027 [arXiv:0904.3304] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  12. G.T. Horowitz and D. Marolf, Quantum probes of space-time singularities, Phys. Rev. D 52 (1995) 5670 [gr-qc/9504028] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  13. M. Blau, D. Frank and S. Weiss, Scalar field probes of power-law space-time singularities, JHEP 08 (2006) 011 [hep-th/0602207] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  14. V.E. Hubeny, M. Rangamani and S.F. Ross, Causal structures and holography, JHEP 07 (2005) 037 [hep-th/0504034] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  15. S. Hawking and G. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge U.K. (1973) [SPIRES].

    Book  MATH  Google Scholar 

  16. E. Minguzzi and M. Sanchez, The causal hierarchy of spacetimes, gr-qc/0609119 [SPIRES].

  17. Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  18. D. Brecher, A. Chamblin and H.S. Reall, AdS/CFT in the infinite momentum frame, Nucl. Phys. B 607 (2001) 155 [hep-th/0012076] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  19. J. Rosen, Embedding of various relativistic Riemannian spaces in pseudo-Euclidean spaces, Rev. Mod. Phys. 37 (1965) 204.

    Article  MATH  ADS  Google Scholar 

  20. C. Collinson, Embeddings of the plane-fronted waves and other space-times, J. Math. Phys. 9 (1968) 403.

    Article  MATH  ADS  Google Scholar 

  21. M. Blau, J.M. Figueroa-O’Farrill and G. Papadopoulos, Penrose limits, supergravity and brane dynamics, Class. Quant. Grav. 19 (2002) 4753 [hep-th/0202111] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. M. Blau and M. O’Loughlin, Homogeneous plane waves, Nucl. Phys. B 654 (2003) 135 [hep-th/0212135] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  23. E. Minguzzi, Chronological spacetimes without lightlike lines are stably causal, Commun. Math. Phys. 288 (2009) 801 [arXiv:0806.0153] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. V.E. Hubeny, M. Rangamani and S.F. Ross, Causal inheritance in plane wave quotients, Phys. Rev. D 69 (2004) 024007 [hep-th/0307257] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  25. P. Hořava and C.M. Melby-Thompson, Anisotropic conformal infinity, arXiv:0909.3841 [SPIRES].

  26. T. Heinzl, Light-cone quantization: Foundations and applications, Lect. Notes Phys. 572 (2001) 55 [hep-th/0008096] [SPIRES].

    Article  ADS  Google Scholar 

  27. S. Hellerman and J. Polchinski, Compactification in the lightlike limit, Phys. Rev. D 59 (1999) 125002 [hep-th/9711037] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  28. P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Ann. Phys. 144 (1982) 249 [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. S. Moroz, Below the Breitenlohner-Freedman bound in the nonrelativistic AdS/CFT correspondence, Phys. Rev. D 81 (2010) 066002 [arXiv:0911.4060] [SPIRES].

    ADS  Google Scholar 

  30. P. Minces and V.O. Rivelles, Energy and the AdS/CFT correspondence, JHEP 12 (2001) 010 [hep-th/0110189] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  31. E.D. Rainville, Special functions, Chelsea Publ. Co., Bronx New York U.S.A. (1971).

    MATH  Google Scholar 

  32. W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and theorems for the special functions of mathematical physics, Third Edition, Springer-Verlag, Berlin Germany (1966).

    MATH  Google Scholar 

  33. A. Bezubik and A. Strasburger, A new form of the spherical expansion of zonal functions and Fourier transforms of SO(d)-finite functions, SIGMA 2 (2006) 033 [math-ph/0603011].

    MathSciNet  Google Scholar 

  34. A. Volovich and C. Wen, Correlation functions in non-relativistic holography, JHEP 05 (2009) 087 [arXiv:0903.2455] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  35. E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS/CFT correspondence, hep-th/0201253 [SPIRES].

  36. R.G. Leigh and N.N. Hoang, Real-time correlators and non-relativistic holography, JHEP 11 (2009) 010 [arXiv:0904.4270] [SPIRES].

    Article  ADS  Google Scholar 

  37. I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  38. D. Marolf and S.F. Ross, Boundary conditions and new dualities: vector fields in AdS/CFT, JHEP 11 (2006) 085 [hep-th/0606113] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  39. J. Moore and R. Schlafly, On equivariant isometric embeddings, Math. Z. 173 (1980) 119.

    Article  MATH  MathSciNet  Google Scholar 

  40. S. Schäfer-Nameki, M. Yamazaki and K. Yoshida, Coset construction for duals of non-relativistic CFTs, JHEP 05 (2009) 038 [arXiv:0903.4245] [SPIRES].

    Article  ADS  Google Scholar 

  41. I.S. Gradshteyn and I.H. Ryzhik, Tables of integrals, series and products, Academic, New York U.S.A. (1980).

    Google Scholar 

  42. U.H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Vacua, propagators and holographic probes in AdS/CFT, JHEP 01 (1999) 002 [hep-th/9812007] [SPIRES].

    Article  ADS  Google Scholar 

  43. G. Compere, S. de Buyl, S. Detournay and K. Yoshida, Asymptotic symmetries of Schrödinger spacetimes, JHEP 10 (2009) 032 [arXiv:0908.1402] [SPIRES].

    Article  ADS  Google Scholar 

  44. J. Hartong, E. Imeroni and B. Rollier, Thermodynamic properties of TsT transformed AdS black holes, work in progress.

Download references

Author information

Authors and Affiliations

  1. Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, Bern University, Sidlerstrasse 5, CH-3012, Bern, Switzerland

    Matthias Blau, Jelle Hartong & Blaise Rollier

Authors
  1. Matthias Blau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jelle Hartong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Blaise Rollier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jelle Hartong.

Additional information

ArXiv ePrint: 1005.0760

Rights and permissions

Reprints and permissions

About this article

Cite this article

Blau, M., Hartong, J. & Rollier, B. Geometry of Schrödinger space-times II: particle and field probes of the causal structure. J. High Energ. Phys. 2010, 69 (2010). https://doi.org/10.1007/JHEP07(2010)069

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2010)069

Keywords

Search

Navigation