The nontriviality of trivial general covariance: How electrons restrict 鈥榯ime鈥?coordinates, spinors (almost) fit into tensor calculus, and of a tetrad is surplus structure

详细信息	查看全文 | 推荐本文 |

• 作者：J. Brian Pitts ^{jpitts@nd.edu}
• 关键词：Electron ; Spinor ; Fermion ; General covariance ; Geometric object ; Nonlinear group representation ; Conventionality of simultaneity ; Schwarzschild solution ; Absolute object
• 刊名：Studies in History and Philosophy of Science Part B ; Studies in History and Philosophy of Modern Physics
• 出版年：2012
• 期刊代码：26_13552198
• 类别：art
• 出版时间：February, 2012
• 卷：43
• 期：1
• 页码：1-24
• 文件大小：483 K

摘要

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors as such cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists鈥?concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or 鈥渢etrad鈥?formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is (nearly) Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but 鈥渢ime鈥?must be listed first. Here 鈥渢ime鈥?is defined in terms of an eigenvalue problem involving the metric components and the matrix diag(鈭?,1,1,1), the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and values of the fields involved.

Apart from coordinate order and the usual spinorial two-valuedness, (densitized) Ogievetsky-Polubarinov spinors form, with the (conformal part of the) metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, six of which are gauged away by a new local O(1,3) gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the nine-component symmetric unimodular square root of the part of the metric that fixes null cones. Thus of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors (related to fermions, with half-integral spin) and tensors and the like (related to bosons, with integral spin) is achieved, such as regarding conservation laws.

Regarding the conventionality of simultaneity, an unusually wide range of values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited.