Reinterpretation, Reformulation and Correction to Adler’s Proposal

Following Adler’s argument, the Riemannian line element ds²_R = g_µνdx^µdx^ν can be dis-cretized and factorized via the measurement of the square of the linear line element

∆s²_A =∆sˆ² = h |γ_µ∆x^µγ_ν∆x^ν| i , (7.1) with

γ^I, γ^J

pq = 2η^IJ ⊗ I_pq, (7.2)

γ^I, γ^J

pq = −2iσ^IJ_pq , (7.3)

γ^µ= e^µ_Iγ^I, (7.4)

g_µν = η_IJe^I_µe^J_ν, η_IJ = g^µνe^I_µe^J_ν , (7.5) where e is the tetrad field and { , } is the anti-commutator.

Care must be taken when one deals with the definition and the interpretation of the measure. In Adler’s approach, Eq.(6.13) was used to define the braket, while Eq.(6.12) was interpreted as a distance functional. However, this choice suffers some drawbacks and is unsuitable for the construction of our quantum spacetime theory.

First, the linear line element operator contains the exact information one would expect to be hidden inside the Hilbert space of the quantized spacetime, i.e., the direction. In the original interpretation, the Hilbert space contains the information for the uncertainty of distance measurement rather than the direction itself. The direction of the line element is provided externally in Eq.(6.12) since the theory is describing a quantized distance functional, rather than a quantized spacetime. The existence of a favored direction that minimizes the uncertainty also breaks Lorentz invariance and isotropy at the smallest

scale. The salient feature of Lorentz symmetry in Adler’s theory (Dirac’s way of taking square root clearly is Lorentz invariant) is therefore lost.

Second, when one measures the proper distance of null eigenstates (which should be quite common given the fact that all particles are massless prior to electro-weak symmetry breaking) along any direction, because of the choice of normalization the proper distance would always be zero. So for a null state even if the curve is not along null direction the proper distance would still be null. To wit, the measure of proper distance is completely uncertain for a non-null displacement on a null state.

Third, the outcome of the linear distance functional depends heavily on the choice of the representation of the Clifford algebra. A complex representation could result in a complex proper distance, which is a radical departure from usual GR. Although Adler tries to address this issue by fixing the representation, the problem still exists as long as the proper distance, being a physical measure, is not a scalar of Clifford algebra, i.e. not representation independent.

Clearly, these observations indicate the necessity of reinterpretation of Adler’s linear line element and a new choice of normalization. We look for new definition that should satisfy Lorentz symmetry, should not have preferred direction, should produce reasonable results for null states, and should be independent of the choice of the representation.

So we give up Eq. (6.13) and introduce a new operator ∆ ˆX^I, called “spacetime interval operator”:

1 , ∀ time-like states 0 , ∀ null states

−1 , ∀ space-like states

(7.8)

Here λ is the characteristic length of the quantized spacetime that is of the order of Planck length and will be derived in sec.7.2, n^I is a non-null vector with nIn^I = ±1, and k^I is a null vector with positive k⁰. The appearance of the γ⁰, and the choice of normalization can be appreciated by looking at the solution of Dirac field equation (See, for example, Ch. 3-3 of Peskin & Schroeder [27].) Physically the insertion of γ⁰makes the normaliza-tion condinormaliza-tion Lorentz invariant, and for the massless case the choice of normalizanormaliza-tion is equivalent to the introduction of the foliation along the time coordinate.

∆ ~X ∆V₃ ∆~V ∆t ∆s² ∆ ~A ∆ ~A_t ∆V₄

∆ ~X O O O O X X X X

∆V3 O O O O X X X X

∆~V O O O O O O O O

∆t O O O O O O O O

∆s² X X O O O O X X

∆ ~A X X O O O O X X

∆ ~A_t X X O O X X O O

∆V4 X X O O X X O O

Table 7.1: A commutativity table showing possible ways of labelling Hilbert space. For elements T_mn inside the table, “O” means m-th basis commutes with n-th basis, and

“X” means non-commutativity. Here all vectors are along spatial eigen-direction nⁱ = h∆Xⁱi, and ∆ ~X = n_i∆Xⁱ is the spatial interval, ∆V₃ = ∆X¹∆X²∆X³ is the time-like 3-volume, ∆~V = −n_iⁱ_jk∆X⁰∆X^j∆X^k is the spatial 3-volume, ∆t is the time difference, ∆s² is the proper distance square, ∆ ~A = niⁱ_jk∆X^j∆X^k is the spatial area,

∆ ~A_t = n_i∆X⁰∆Xⁱ is the time-like area, and ∆V₄ = ∆X⁰∆X¹∆X²∆X³ is the 4-volume. Notice that actually ~V can always be described by products of two non-trivial quantum numbers in the system.

This new measurement is not a distance measure at all, but a local spacetime interval operation on the exponential map of Riemann Normal Coordinate. One should not misin-terpret the operator as a coordinate difference operator, since it actually lies on the tangent bundle of the manifold. A better way to understand it is to treat it as a discretized version of the velocity 4-vector. And only when combined with the tetrad does the coordinate difference measure ∆ ˆX^µreappear.

Just like the components of a vector in GR, ∆ ˆX^I (gamma matrices) are not physical objects. To measure the velocity of a particle we need two-particle interaction, and the physical object is the inner product rI∆ ˆX^I, where rIis the classical trajectory of a probe particle expressed in the same representation as ∆ ˆX^I. No matter what representation of Clifford algebra we are choosing, the measurement is always a scalar. Therefore all the derivations and results we obtained are representation independent. The only exception is that in sec. 6 we require the realness of ∆ ˆX^I during derivation of GUP. However one should get the same GUP regardless of the representation used.

The new choice of interpretation also implies the existence of an underlying minimal distance. Due to the special structure of Clifford algebra, one can immediately obtain ∆X^I∆X^I

 = λ² for non-null cases and 0 for null cases, implying that there are un-certainties within the spacetime interval measurement similar to what was obtained in Ref.[22], where the uncertainty lies on proper distance measurement. But in our case even such uncertainties are Lorentz invariant. One may try to obtain the variance of inter-val measure: along direction there is no uncertainty at all since it is the direction where eigen-states are defined. However along the transverse direction the measurement is completely uncertain. From this point of view the behaviour of the spacetime interval operator is exactly the same as spin operators in relativistic QM. They both have 3 definite quantum numbers, i.e., S²/∆t, ~S/∆ ~X along spatial eigen-direction, and helicity/∆V₃, where ∆V₃ is the spatial 3-volume. There are also other possible ways of labelling Hilbert space, which are shown in Table 7.1.

Na¨ıvely, in SO(3, 1) system there are 4 quantum numbers: the 4-momentum. How-ever they do not correspond to the quantum numbers in our theory since the momentum

operators do not commute with each other. Only at the decoherence limit will the ad-ditional 4 quantities, eigen-direction, spatial 3-volume along eigen-direction and proper distance, emerge.

An important feature of Adler’s original interpretation is that the proper time differ-ence ∆s ∝ γ^µcan have two eigenvalues of same magnitude but opposite sign. Therefore in his theory it is permissible to move backward in time, zigzag around the same spot at high temperature if no rule forbids the excitation of these states. However in our inter-pretation, time difference is now proportional to the identity operator due to the choice of measure, rendering its expectation value positive definite. Thus arrow of time problem is perfectly solved without invoking second law of thermodynamics.

In Ref.[22], Adler suggests that one may take these tiny line elements as building blocks of a curve, without specifying what kind of curve it is. Since in the original paper the measurement of linear line element operators are associated with SO(3, 1) proper distance, this curve resembles a geodesic, called “quantized geodesic” due to its discrete nature. Under our spacetime interval operators, the link to the geodesic becomes more explicit since a trajectory is specified along the geodesic as

|Ci = ⊕

Here |Ci is the composite state for geodesic, and (i) indicates the i-th site along the geodesic. Following usual convention in special relativity, positive eigenvalue of proper distance ∆s = ±√

∆s² is chosen due to the positivity of time difference.