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**Contents:**

The formula as such does however not contain any information about the state of the system; it would evaluate to the same operator regardless of what state the system is in. Quantum fields relate to quantum mechanics as classical fields do to classical mechanics, i. In particular, the quantum fields are not wavefunctions, even though the equations which govern their time evolution may be deceptively similar to those of the corresponding wavefunction in a semiclassical formulation.

There is no variation in strength of the fields between different points in spacetime; the variation that happens is rather one of phase factors. Mathematically it may look as though all of the fields are vector-valued in addition to being operator-valued , since they all have several components, can be multiplied by matrices, etc. The B, W j , and G a fields are all vectors in this sense, so the corresponding particles are said to be vector bosons.

Therefore, these constitute a third kind of quantity, which is known as a spinor. As is common in quantum theory, there is more than one way to look at things. At first the basic fields given above may not seem to correspond well with the "fundamental particles" in the chart above, but there are several alternative presentations which, in particular contexts, may be more appropriate than those that are given above.

Each of these is a four component bispinor , for a total of 96 complex-valued components for the fermion field. This is very important in the Standard Model because left and right chirality components are treated differently by the gauge interactions. Put more simply, the weak interaction could rotate e. This does not however change the experimentally-proven chiral nature of the weak interaction. A distinction can thus be made between, for example, the mass and interaction eigenstates of the neutrino. The former is the state which propagates in free space, whereas the latter is the different state that participates in interactions.

Which is the "fundamental" particle? We can switch between these states using the CKM matrix for the quarks, or the PMNS matrix for the neutrinos the charged leptons on the other hand are eigenstates of both mass and flavour. As an aside, if a complex phase term exists within either of these matrices, it will give rise to direct CP violation , which could explain the dominance of matter over antimatter in our current universe.

This is not so common when a quantum field theory has been set up, but often features prominently in the process of quantizing a field theory. To retain gauge invariance, the underlying fields must be massless, but the observable states can gain masses in the process. These states are:. The Z field actually contributes in every process the photon does, but due to its large mass, the contribution is usually negligible. Much of the qualitative descriptions of the standard model in terms of "particles" and "forces" comes from the perturbative quantum field theory view of the model.

The free fields care for particles in isolation, whereas processes involving several particles arise through interactions. The idea is that the state vector should only change when particles interact, meaning a free particle is one whose quantum state is constant. This corresponds to the interaction picture in quantum mechanics. In the alternative Heisenberg picture , state vectors are kept constant, at the price of having the operators in particular the observables be time-dependent.

The interaction picture constitutes an intermediate between the two, where some time dependence is placed in the operators the quantum fields and some in the state vector. In QFT, the former is called the free field part of the model, and the latter is called the interaction part. The free field model can be solved exactly, and then the solutions to the full model can be expressed as perturbations of the free field solutions, for example using the Dyson series.

It should be observed that the decomposition into free fields and interactions is in principle arbitrary. For example, renormalization in QED modifies the mass of the free field electron to match that of a physical electron with an electromagnetic field , and will in doing so add a term to the free field Lagrangian which must be cancelled by a counterterm in the interaction Lagrangian, that then shows up as a two-line vertex in the Feynman diagrams.

This is also how the Higgs field is thought to give particles mass : the part of the interaction term which corresponds to the nonzero vacuum expectation value of the Higgs field is moved from the interaction to the free field Lagrangian, where it looks just like a mass term having nothing to do with Higgs. These equations can be solved exactly. In the periodic case, the solution for a field F any of the above can be expressed as a Fourier series of the form. For these derivations, one starts out with expressions for the operators in terms of the quantum fields.

An important step in preparation for calculating in perturbative quantum field theory is to separate the "operator" factors a and b above from their corresponding vector or spinor factors u and v.

We call in Fractal Physics, the Metrics of the 5th Dimension, the Generator of the Universe since it generates the scales, topologies and forms of reality - a simple. Meet the mathematical masters of the universe. It also uses random number generator equations for timing signals, trigonometric equations.

The vertices of Feynman graphs come from the way that u and v from different factors in the interaction Lagrangian fit together, whereas the edges come from the way that the a s and b s must be moved around in order to put terms in the Dyson series on normal form. The Lagrangian can also be derived without using creation and annihilation operators the "canonical" formalism , by using a "path integral" approach, pioneered by Feynman building on the earlier work of Dirac.

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See e. Path integral formulation on Wikipedia or A.

Zee's QFT in a nutshell. This is one possible way that the Feynman diagrams , which are pictorial representations of interaction terms, can be derived relatively easily. A quick derivation is indeed presented at the article on Feynman diagrams. We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model Lagrangian density. Any such term must be both gauge and reference-frame invariant, otherwise the laws of physics would depend on an arbitrary choice or the frame of an observer. The three factors of the gauge symmetry together give rise to the three fundamental interactions, after some appropriate relations have been defined, as we shall see.

A complete formulation of the Standard Model Lagrangian with all the terms written together can be found e. A free particle can be represented by a mass term, and a kinetic term which relates to the "motion" of the fields. Generally, as below, this term is included within the couplings creating an overall "dynamical" term. For the spin-1 fields, first define the field strength tensor.

In an Abelian commutative group such as the U 1 we use here , since the generators t a all commute with each other, the structure constants vanish. The two-index objects are the field strengths derived from W and G the vector fields. There are also two extra hidden parameters: the theta angles for SU 2 and SU 3. Note that we have to redefine a new U 1 symmetry of weak hypercharge , different from QED, in order to achieve the unification with the weak force.

The first convention, used in this article, is equivalent to the earlier Gell-Mann—Nishijima formula. It makes the hypercharge be twice the average charge of a given isomultiplet. One may then define the conserved current for weak isospin as. As explained above , these currents mix to create the physically observed bosons, which also leads to testable relations between the coupling constants. To explain this in a simpler way, we can see the effect of the electroweak interaction by picking out terms from the Lagrangian. The U 1 symmetry, on the other hand, is similar to electromagnetism, but acts on all " weak hypercharged " fermions both left- and right-handed via the neutral Z 0 , as well as the charged fermions via the photon.

The quantum chromodynamics QCD sector defines the interactions between quarks and gluons , with SU 3 symmetry, generated by T a. Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by. We see that the mass-generating interaction is achieved by constant flipping of particle chirality. The physicists encountered repeated failures while trying to construct a machine which could generate the infinite improbability field needed to flip a spaceship across the mind-paralyzing distances between the farthest stars.

They eventually announced that such a machine was virtually impossible. Then, one day, a student who had been left to sweep up after a particularly unsuccessful party found himself reasoning in this way: If he thought to himself, such a machine is a virtual impossibility, it must have finite improbability.

So all I have to do in order to make one is to work out how exactly improbable it is, feed that figure into the finite improbability generator, give it a fresh cup of really hot tea The Infinite Improbability Drive is featured in every form of Hitchhiker's media, which includes the following:. The set of all QVPs with equal physical status is given by. Consider first the situation where there is only one generator of translations over the generic axis W , i.

The one-generator situation occurs for the one-dimensional spatial case where corresponds to the momentum operator in the x direction, say. The one-generator situation also occurs for QVPs over time in the case of T symmetry where corresponds to , and hence an equivalent situation results. Two comments are in order here. There is also no equation of motion inherent in the theory to represent time evolution of the galaxy over extended periods of time. The study of states that are localized in time like this has a history dating back to Stueckelberg in the s and Feynman, Nambu and Schwinger in the s [ 23 ].

However, in contrast with that here, these previous studies incorporated dynamical equations as an integral part of their approaches and inherently assumed time evolution. Next, consider the situation where there are two distinct generators, i. This occurs for QVPs over time when T violation is present, i.

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We assume the commutator of the two versions of the Hamiltonians to be of the simple form. The operators and are the corresponding weighted averages of and and represent the phenomenological Hamiltonians that would be observed within the galaxy. The parameter t c represents the corresponding net translation in time that would be observed within the galaxy using conventional clocks.

Bi-directional time evolution like this has been explored previously by Carroll, Barbour and co-workers [ 24 , 25 ]. This implies that if both clocks are synchronized at one particular value of t c , they will remain in synchrony for larger values of t c. We assume all clocks, irrespective of where they are located, are synchronized to display t c , which hereafter we refer to as the clock time. Also, for brevity, the location of the galaxy will only be quoted explicitly for positive values of time, leaving the corresponding situation for negative values implicit, unless clarity warrants otherwise.

All of these results taken together show that the new theory offers a framework that gives the origin of dynamics. We are now in a position where we can interpret the new theory in terms of the block universe. For this, we restrict our attention to T violation and the temporal case.

In this respect, the new theory is consistent with the principles of the block universe view. The new theory, however, includes an additional feature. The direction of time in the conventional block universe is the one associated with increasing entropy. Conventionally, only time-symmetric laws are considered, and hence any time asymmetry in entropy is necessarily associated with asymmetric temporal boundary conditions, i.

As a result, the new theory allows the block universe view to be extended, accordingly, to one containing an inherent direction of time. The new theory offers some support for our subjective notions about time. In other words, any particular clock time value t c is associated with an ordered collection of conditional states. Finally, there is also a severe constraint for us. The new theory is deterministic in the sense that it obeys Hamiltonian dynamics and hence it does not allow us to act freely nor to have an ability to determine the future, despite our strong subjective perceptions to the contrary.

In short, it offers us no escape from determinism. Quantum mechanics has spawned the development of a myriad of advanced technologies that drive the modern western way of life. However, the foundations of quantum mechanics offer something more, a reason to question how we view ourselves within an objective landscape. In this paper, we explored the implications of the quantum nature of time for humans. An important feature of the new theory is that dynamics and time evolution are not assumed to be part of its structure, rather they emerge phenomenologically due to T violation.

The state of a physical system over time is given by a collection of conditional states, where each conditional state represents the system at a specific value of clock time. We found the new theory is consistent with the block universe in that the conditional states are fixed by unitary evolution and that no one conditional state can be singled out as representing the present, and divide other conditional states into sets representing the past and the future.

However, the new theory has additional features.

One is that each conditional state is localized in time around a mean clock time value and hence it can represent a human having the subjective perception of the present moment. Another is that the collection of conditional states has a natural ordering in terms of the degree of time evolution, and as a result, each conditional state contains, in general, evidence of past evolution. This supports the subjective perception of being able to recall the past and contemplate the future.

It also provides an objective basis for the subjective perception of a flow or passage of time. In short, the collection of conditional states can represent a human at every clock time as having the subjective perception of the present moment with memory of the past and awareness of the passage of time. These results extend the block universe view. The proof of the theorem hinges on the partitioning of the universe into two distinct parts and assuming each part has a distinct physical description.

However, this is not the whole story [ 7 , 31 ]: a violation of the theorem implies the logical complement of the assumptions it hinges on, namely that the LR—MS partitioning is not respected by nature, and hence either the MS part does not give settings that are independent of the LR part, or the LR part does not have a local realistic description. Here, we explore the specific case, as imagined by Bell and others [ 5 , 37 , 39 — 41 ], where the measurement settings are determined by experimentalists in the MS part.

In this case, the implications of a violation is either. We have no physical grounds to favour either i or ii and hence it is up to us to choose which possibility we prefer. However, if we insist on the view that humans can act independently of the universe that surrounds them, then the only compatible possibility is ii. The question of which interpretation is preferable has a precedent in the debate about celestial motion in the s as follows. The Ptolemaic model of the s is analogous to the leading interpretation of a Bell theorem violation now: both give humans a preferred status i.

Alternatively, the Copernican model in the s is analogous to the fully causal interpretation: both eliminate the preferred status of humans and both account for the unexplained phenomena in a simple way. For example, Bell et al. Specifically, arguing for experimenter—subject independence on the basis that the alternative has undesirable consequences does not prove that experimenters are independent of their subjects. Rather, the alternative may well be true, in which case we would need to deal with the consequences. For example, imagine the experimenters eliminate all sources of experimenter—subject dependence possible and they perform experiments and analyse results in an accepted scientific manner.

Then imagine a fundamental kind of experimenter—subject dependence is discovered that cannot be eliminated and, furthermore, it has occurred in all experiments to date and will occur in all future experiments. With no way to improve the experiments, there is no reason to reject the results of science to date, nor to abandon practising science in the way it was done.

The consequences of the alternative to experimenter—subject independence would simply be the conditions under which science is done. Indeed, these conditions include various conservation laws. In particular, Unnikrishnan has shown that Bell inequalities associated with correlated spin particles are not consistent with the conservation of spin angular momentum, and that satisfying this conservation law requires the Bell inequalities to be violated by an amount given by quantum mechanics [ 42 — 44 ].

What Bell et al. In particular, the constraint of bounded momentum restricts spatial distributions to having a minimum width, but allows a position eigenstate basis expansion. We are associating here all elements with the present, and each element has evidence of prior evolution.

National Center for Biotechnology Information , U. Published online May Joan A. Author information Article notes Copyright and License information Disclaimer. Accepted Apr 4. This article has been cited by other articles in PMC. Abstract Advances in our understanding of the physical universe have dramatically affected how we view ourselves.

Keywords: time, violation of time reversal symmetry, arrow of time. Introduction Advances in science have had a profound impact on how we view ourselves. Accordingly, we begin with a representation of the quantum state of a system that treats time and space symmetrically by adopting the following: Principle 2. Principle 2. Consistency with the block universe We are now in a position where we can interpret the new theory in terms of the block universe.

Implications for humans The new theory offers some support for our subjective notions about time. Conclusion Quantum mechanics has spawned the development of a myriad of advanced technologies that drive the modern western way of life. Appendix A. Footnotes 1 For convenience, we use units for which.