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Physics LibreTexts

1.2: Comments on the Poincare Algebra

  • Page ID
    9384
  • l. Lorentz invariant physical system

    A Lorentz invariant physical system is defined and discussed in some detail in Chapter 3. Examples of Lorentz inva¡iant physical systems are given in QLB: Some Lorentz Invariant Systems.

    2. Poincare Algebra: the basic equations of physics

    (2.1) to (2.9) are the Poincare Algebra. The Poircare Algebra is the Lie Algebla for the Poincare group.

    The appearance of Planck's constant \(h\) and the speed of light \(c\) in (2.6) and (2.9) indicates explicitly that the Poincare Algebra involves quantum mechanics and special relativity.

    The Poincare Algebra was frst derived i¡ the 1930's. The central role it plays in relativistic quantum mechanics is emphasized in Dirac (1949).

    The corresponding equations for a Galilei invariant (or nonrelativistic) physical system, the Galilei Algebra, are given in item 8.

    (2.1) to (2.9) are the basic equations of relativistic quantum mechanics. Indeed, since non¡elativistic quantum mechanics and classical mechanics a¡e approximations to relativistic quantum mechanics, (2.1) lo (2.9) are the basic equations of physics.

    3. Poincare generators

    The ten Hermitian operators H,Pi,Ji,,äi are the Poinca¡e generators for the physical system.

    IJ is the Hamiltonian for the system.

    ¡,i , .7 i , Iil are the jth component of the total momentum, total angular momentum and Lorentz booster, respectively, for the system.

    As discussed in Chapter 3, these operators generate time translations, spatial displacements, rotations and Lorentz boosts.

    4. Symmetry in the Poincare Algebra

    (2.1) tt¡ (2.9) a¡e invariant under the replacement H,I(r '-- -H,-Iir (2.10) With one exception, the Hamiltonians in the examples in QLB: Some I'orentz Invariant Systems arc chosen to have positive spectral values.

    The Hamiltonian for the Dirac particle discussed in QI'B: Some Lorentz Invariant Systems is the one exception: it has both positive and negative spectral values. The negative energy states of the Dirac particle have no physical interpretation in a one-particle theory. We outline in QLB: Some Lorentz Invariant Systems how Dirac's brilliant interpretation of these states in 1930 predicted the existence of antiparticles and led to the invention of relativistic quantum field theory.

    5. Constants of the motion

    (2.2) and (2.4) show that ail components of the total momentum and the total angular momentum are constants of the motion. It is often convenient to desc¡ibe a system using eigenkets of the total momentum and eigenvectors of the total angulal momentum.

    (2.7) shows that none of the components of the Lorentz booster are constants of the motion. It is generally not convenient to desc¡ibe a system using the eigenkets of the Lorentz boosters.

    Equations (2.1) to (2.5): the equations not involving \(K^j}

    These equations a¡e derived without involvin g Lorentz boosts. They are the same in nomelativistic and relativistic quantum mechanics. (See item 8.) Equations (2.6) to (2.9): the equations involvinq 1(j These equations involve ths speed of light c. 'I'lre equations are in two pairs: one pair couples H and Pi and the other couples Jr and I{i. We show in Chapter 3 that the coupling of H a¡d pi yields a mixing of energy and momentum under a Lorentz boost. This mixing is familiar from classical mechanics. We show in Chapter 6 that the coupling of Jj and ¡-j yields a Wigner rotation of internal angular momentum under a Lorentz boost.

    The nonrelativistic Limit: the Galilei Algebra

    The Galilei Algebra is a set of communtation relations appropriate for describing a Galilei invariant physical system. The Galilei Algebra involves the Galilei generators which are the Hamiitonian and tho three components of the total momentum, total angular momentum and Galilei booster for the system. As discussed in Chapter 3 these operators generate time translations, spatial displacements, rotations and Galilei boosts.

    Since non¡elativistic quantum mechanics is the special case of relativistic quantum mechanics corresponding to taking the speed of light c to be infinite, the Galilei Algebra is identical to the Poincare Algebra except for (2.6) and (2.9). These are the only equations in the Poincare Algebra which involve c.

    We show in Chapter 3 that the Galilei Algebra differs from the Poincare Algebra only in having (2.6) and (2.9) replaced by 8 [r.r, "o] : -irtm6¡n lr