| Summary: | We consider the theory of quantum gravity in which gravity emerges as a result of the symmetry-breaking transition in the quantum vacuum. The gravitational tetrads, which play the role of the order parameter in this transition, are represented by the bilinear combinations of the fermionic fields. In this quantum gravity scenario the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mi>s</mi></mrow></semantics></math></inline-formula> in the emergent general relativity is dimensionless. Several other approaches to quantum gravity, including the model of superplastic vacuum and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>F</mi></mrow></semantics></math></inline-formula> theories of gravity support this suggestion. The important consequence of such metric dimension is that all the diffeomorphism invariant quantities are dimensionless for any dimension of spacetime. These include the action <i>S</i>, cosmological constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Λ</mi></semantics></math></inline-formula>, scalar curvature <i>R</i>, scalar field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Φ</mi></semantics></math></inline-formula>, wave function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>, etc. The composite fermion approach to quantum gravity suggests that the Planck constant <i>ℏ</i> can be the parameter of the Minkowski metric. Here, we extend this suggestion by introducing two Planck constants, bar <i>ℏ</i> and slash <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>/</mo><mspace width="-0.08em"></mspace><mspace width="-0.166667em"></mspace><mspace width="-0.166667em"></mspace><mi>h</mi></mrow></semantics></math></inline-formula>, which are the parameters of the correspondingly time component and space component of the Minkowski metric, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>g</mi><mi>Mink</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><mi>diag</mi><mrow><mo>(</mo><mo>−</mo><msup><mo>ℏ</mo><mn>2</mn></msup><mo>,</mo><mo>/</mo><mspace width="-0.08em"></mspace><mspace width="-0.166667em"></mspace><mspace width="-0.166667em"></mspace><msup><mi>h</mi><mn>2</mn></msup><mo>,</mo><mo>/</mo><mspace width="-0.08em"></mspace><mspace width="-0.166667em"></mspace><mspace width="-0.166667em"></mspace><msup><mi>h</mi><mn>2</mn></msup><mo>,</mo><mo>/</mo><mspace width="-0.08em"></mspace><mspace width="-0.166667em"></mspace><mspace width="-0.166667em"></mspace><msup><mi>h</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The parameters bar <i>ℏ</i> and slash <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>/</mo><mspace width="-0.08em"></mspace><mspace width="-0.166667em"></mspace><mspace width="-0.166667em"></mspace><mi>h</mi></mrow></semantics></math></inline-formula> are invariant only under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> transformations, and, thus, they are not diffeomorphism invariant. As a result they have non-zero dimensions—the dimension of time for <i>ℏ</i> and dimension of length for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>/</mo><mspace width="-0.08em"></mspace><mspace width="-0.166667em"></mspace><mspace width="-0.166667em"></mspace><mi>h</mi></mrow></semantics></math></inline-formula>. Then, according to the Weinberg criterion, these parameters are not fundamental and may vary. In particular, they may depend on the Hubble parameter in the expanding Universe. They also change sign at the topological domain walls resulting from the symmetry breaking.
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