PRL 114, 058902 (2015)
PHYSICAL
REVIEW
Comment on “Quantum Frameness for CPT Symmetry” Skotiniotis et al. [1] deal with the problem of the definition of CPT symmetry, which would be useful in the framework of quantum information theory. Beyond any doubt, one can say that CPT symmetry [2-6] is one of the cornerstones of quantum physics. Therefore, much care must be exercised when attempting to modify the standard approach to this problem. In the case of the abovementioned paper, neither the geometric meaning of CPT transformation nor the CPT theorem are retained. The operator considered by Skotiniotis et al. is merely some simple unitary operator which is a symmetry of a free particle theory but will be, in general, broken when interaction is switched on. It is not useful beyond the scope of kinematics of free particles. The CPT symmetry and CPT theorem have a universal character due to the following reasons. First, it predicts that for any particle there exists the corresponding antiparticle of opposite charge, the same mass, spin, and lifetime; this conclusion is confirmed, with no exception, by the experi ment. It is important to note that, in principle, any kind of interaction contributes to the particle properties. Success of the predictions based on CPT symmetry comes from the fact that it is not broken by any interaction [7]. The celebrated CPT theorem states that any quantum theory which is Poincare invariant and causal (in other words, obeys the local commutativity condition as described by the spin-statistics relation) enjoys CPT symmetry. The impor tance of the CPT theorem cannot be overestimated: once the covariance and causality are assumed, CPT symmetry is their unavoidable consequence for interactions pre serving Lorentz invariance and locality. Now comes the important point. In order to define physically acceptable interactions (clustering, causality, etc.) one appeals to the space-time description (one could work exclusively in momentum space but then the relevant conditions would be highly nonlocal and not intuitive) [8], The proof of the CPT theorem relies heavily on the natural interpretation of CPT transformation in terms of space-time variables; in particular, it must involve time inversion (this is why CP is not a universal symmetry). To encode this property, one does not have to appeal directly to the description in terms of space-time variables; it is sufficient to impose the appropriate algebraic relations with Poincare generators. In particular, the CPT operator should anti commute with the tim e translation generator multiplied by imaginary unit i. As a result, the CPT symmetry can be reconciled with the positivity of energy (which generates
0031 -9007/15/114(5)/058902( 1)
LETTERS
week ending 6 FEBRUARY 2015
time translations) only provided the CPT operator is antiunitary. The authors of the Letter under consideration define the unitary representation of the Z2 group consisting of unity and, what they call, the CPT operator. It acts in the direct sum of two (particle + antiparticle) irreducible rep resentations of the Poincare group of the same mass and spin and transforms particles into antiparticles of opposite momentum and spin projection; a good evidence that we are not dealing with a CPT operation is the reversal of a particles’ three-momenta which, on the contrary, should be invariant under the action of a properly defined CPT operation. The energy is preserved so, being unitary, the CPT operator commutes with the time translations. Such a CPT operator does not involve time inversion. As a result, switching on interaction will, in general, break this symmetry. A further comment is in order. In their construction, the authors refer to the relevant wave equations. This is completely unnecessary in the noninteracting case. The wave equations are nowhere really used. Concluding, the authors developed a theory of frameness resources related to some simple discrete symmetry of relativistic kinematics which is not the CPT symmetry. P. Kosinski Department of Computer Science Faculty of Physics and Applied Informatics University of L6dz Pomorska 149/153, 90-236 Lodz, Poland Received 6 June 2014; published 5 February 2015 DOI: 10.1103/PhysRevLett. 114.058902 PACS numbers: 03.67.Hk, 11.30.Er, 11.30.Fs [email protected] [1] M. Skotiniotis, B. Toloui, J. T. Durham, and B .C . Sanders, Phys. Rev. Lett. Ill, 020504 (2013); Phys. Rev. A 90, 012326 (2014). [2] J. Schwinger, Phys. Rev. 82, 914 (1951). [3] G. Liiders, Dan. Mat. Fys. Medd. 28, 1 (1954). [4] W. Pauli, in Niels Bohr and the Development o f Physics (Pergamon, New York, 1955). [5] J. S. Bell, Proc. R. Soc. Edinburgh, Sect. A 231, 479 (1955). [6] R. Streater and A. S. Wightmann, PCT, Spin and Statistics and All That (Benjamin, New York, 1964). [7] Some attempts to avoid the CPT theorem are discussed in the literature but this is usually related with the breaking of Lorentz invariance. [8] S. Weinberg, The Quantum Theory o f Fields (Cambridge University Press, New York, 1995), Vol. I.
058902-1
© 2015 American Physical Society
Copyright of Physical Review Letters is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
PHYSICAL
REVIEW
Comment on “Quantum Frameness for CPT Symmetry” Skotiniotis et al. [1] deal with the problem of the definition of CPT symmetry, which would be useful in the framework of quantum information theory. Beyond any doubt, one can say that CPT symmetry [2-6] is one of the cornerstones of quantum physics. Therefore, much care must be exercised when attempting to modify the standard approach to this problem. In the case of the abovementioned paper, neither the geometric meaning of CPT transformation nor the CPT theorem are retained. The operator considered by Skotiniotis et al. is merely some simple unitary operator which is a symmetry of a free particle theory but will be, in general, broken when interaction is switched on. It is not useful beyond the scope of kinematics of free particles. The CPT symmetry and CPT theorem have a universal character due to the following reasons. First, it predicts that for any particle there exists the corresponding antiparticle of opposite charge, the same mass, spin, and lifetime; this conclusion is confirmed, with no exception, by the experi ment. It is important to note that, in principle, any kind of interaction contributes to the particle properties. Success of the predictions based on CPT symmetry comes from the fact that it is not broken by any interaction [7]. The celebrated CPT theorem states that any quantum theory which is Poincare invariant and causal (in other words, obeys the local commutativity condition as described by the spin-statistics relation) enjoys CPT symmetry. The impor tance of the CPT theorem cannot be overestimated: once the covariance and causality are assumed, CPT symmetry is their unavoidable consequence for interactions pre serving Lorentz invariance and locality. Now comes the important point. In order to define physically acceptable interactions (clustering, causality, etc.) one appeals to the space-time description (one could work exclusively in momentum space but then the relevant conditions would be highly nonlocal and not intuitive) [8], The proof of the CPT theorem relies heavily on the natural interpretation of CPT transformation in terms of space-time variables; in particular, it must involve time inversion (this is why CP is not a universal symmetry). To encode this property, one does not have to appeal directly to the description in terms of space-time variables; it is sufficient to impose the appropriate algebraic relations with Poincare generators. In particular, the CPT operator should anti commute with the tim e translation generator multiplied by imaginary unit i. As a result, the CPT symmetry can be reconciled with the positivity of energy (which generates
0031 -9007/15/114(5)/058902( 1)
LETTERS
week ending 6 FEBRUARY 2015
time translations) only provided the CPT operator is antiunitary. The authors of the Letter under consideration define the unitary representation of the Z2 group consisting of unity and, what they call, the CPT operator. It acts in the direct sum of two (particle + antiparticle) irreducible rep resentations of the Poincare group of the same mass and spin and transforms particles into antiparticles of opposite momentum and spin projection; a good evidence that we are not dealing with a CPT operation is the reversal of a particles’ three-momenta which, on the contrary, should be invariant under the action of a properly defined CPT operation. The energy is preserved so, being unitary, the CPT operator commutes with the time translations. Such a CPT operator does not involve time inversion. As a result, switching on interaction will, in general, break this symmetry. A further comment is in order. In their construction, the authors refer to the relevant wave equations. This is completely unnecessary in the noninteracting case. The wave equations are nowhere really used. Concluding, the authors developed a theory of frameness resources related to some simple discrete symmetry of relativistic kinematics which is not the CPT symmetry. P. Kosinski Department of Computer Science Faculty of Physics and Applied Informatics University of L6dz Pomorska 149/153, 90-236 Lodz, Poland Received 6 June 2014; published 5 February 2015 DOI: 10.1103/PhysRevLett. 114.058902 PACS numbers: 03.67.Hk, 11.30.Er, 11.30.Fs [email protected] [1] M. Skotiniotis, B. Toloui, J. T. Durham, and B .C . Sanders, Phys. Rev. Lett. Ill, 020504 (2013); Phys. Rev. A 90, 012326 (2014). [2] J. Schwinger, Phys. Rev. 82, 914 (1951). [3] G. Liiders, Dan. Mat. Fys. Medd. 28, 1 (1954). [4] W. Pauli, in Niels Bohr and the Development o f Physics (Pergamon, New York, 1955). [5] J. S. Bell, Proc. R. Soc. Edinburgh, Sect. A 231, 479 (1955). [6] R. Streater and A. S. Wightmann, PCT, Spin and Statistics and All That (Benjamin, New York, 1964). [7] Some attempts to avoid the CPT theorem are discussed in the literature but this is usually related with the breaking of Lorentz invariance. [8] S. Weinberg, The Quantum Theory o f Fields (Cambridge University Press, New York, 1995), Vol. I.
058902-1
© 2015 American Physical Society
Copyright of Physical Review Letters is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
