User:Michael C Price/draft

Draft page. For draft work and formula library.

Afshar/Azeri
3rr: 4th edit Warned after 3 reverts 3rd edit 2nd edit 1st edit

Semi-protect. High level of disruptive IP POV tag teaming over many months. Last 24 hrs has seen 3 IPs at work. Repeated attempts at dialog with IP on talk page fail. --Michael C. Price talk 10:20, 10 November 2009 (UTC)

at

Essays
WP:ESCA

GR

 * $$S=\int d^4x\sqrt{-g} \; (\frac{R-2\Lambda }{16\pi G} + \mathcal{L}_\mathrm{M})$$

ref format
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Result assumption
Peter J. Lewis (2007). “How Bohm’s Theory Solves the Measurement Problem”, Philosophy of Science 74 (5): 749–760 Lewis on Wallace & Brown's identification of Bohm's result assumption Also Lewis “Empty Waves in Bohmian Quantum Mechanics”, British Journal for the Philosophy of Science 58: 787–803 (2007).

Bohm
Q: Hugh Everett says that Bohm's particles are not observable entities, but surely they are - what hits the detectors and causes flashes?

A:

Occam's razor criticism
Both Hugh Everett III and Bohm treated the wavefunction as a complex-valued but real field. Everett's many-worlds interpretation is an attempt to demonstrate that the wavefunction alone is sufficient to account for all our observations. When we see the particle detectors flash or hear the click of a Geiger counter or whatever then Everett's theory interprets this as our wavefunction responding to changes in the detector's wavefunction, which is responding in turn to the passage of another wavefunction (which we think of as a "particle", but is actually just another wave-packet). But no particle, in the Bohm sense of having a defined position and velocity, is involved in measurement. For this reason Everett sometimes referred to his approach as the "pure wave theory". Talking of Bohm's 1952 approach, Everett says: Our main criticism of this view is on the grounds of simplicity - if one desires to hold the view that $\psi$ is a real field then the associated particle is superfluous since, as we have endeavored to illustrate, the pure wave theory is itself satisfactory.

In the Everettian view, then, the Bohm particles are unobservable entities, similar to, and equally as unnecessary as, for example, the luminiferous ether was found to be unnecessary in special relativity. In Everett's view, we can remove the particles from Bohm's theory and still account for all our observations. The unobservability of the "hidden particles" stems from an asymmetry in the causal structure of the theory; the particles are influenced by a "force" exerted by the wavefunction and by each other, but the particles do not influence the time development of the wavefunction (i.e. there is no analogue of Newton's third law -- the particles do not react back onto the wavefunction ) Thus, if we regard the wavefunction as real and the source of all experience, the particles do not make their presence known in any way; as the theory says, they are hidden, but in a far more profound way than de Broglie and Bohm had intended.

In the Everettian view the role of the Bohm particle is to tag, or select, just one branch of the universal wavefunction; the other branches are designated "empty" and implicitly assumed by Bohm, in what is called the "result assumption", to be devoid of conscious observers. H. Dieter Zeh comments on these "empty" branches:It is usually overlooked that Bohm’s theory contains the same “many worlds” of dynamically separate branches as the Everett interpretation (now regarded as “empty” wave components), since it is based on precisely the same. . . global wave function. .. David Deutsch has expressed the same point more "acerbically" : pilot-wave theories are parallel-universe theories in a state of chronic denial.

This argument of Everett's is sometimes called the "redundancy argument", since the superfluous particles are redundant in the sense of Occam's razor. .

This conclusion has been challenged by pilot wave advocates, with a number of suggested resolutions; either make the "result assumption" explicit, deny that the wavefunction is as objectively real as the particles or dispute whether the Everett prescription is complete (e.g. can probabilities be derived from the wavefunction?)

Wallace and Brown
(abstract, page 1)
 * de Broglie-Bohm theory does have the resources to provide a coherent solution of the measurement problem, but they do not involve the hypothetical corpuscles whose existence is precisely what distinguishes the theory from the Everettian picture of quantum reality.

(page 5)W&B's result assumption, from Bohm part II:
 * It is useful to consider of the precise wording of this [result] assumption.
 * Now, the packet entered by the apparatus [hidden] variable . . . determines the actual result of the measurement, which the observer will obtain when he looks at the apparatus.

(page 6) W&B's question:
 * The crucial question we wish to raise is this. Does this wavepacket, in and of itself, account for the result of the measurement, or does a definite measurement outcome require, even in this case of complete predictability, the presence of the hidden variables within it?

(page 6/7)
 * The Result Assumption appears to be inconsistent with .. or at least to override it in some mysterious way.

(page 7)
 * Quantum mechanics, in our view, has both substance and form before the introduction of hidden corpuscles. In fact, we believe the case has already been made elsewhere, the rest of this paper being an attempt to summarise the bones of the argument and to add a bit more new flesh to it.

(page 8/9)
 * the corpuscle’s role is minimal indeed: it is in danger of being relegated to the role of a mere epiphenomenal ‘pointer’, irrelevantly picking out one of the many branches defined by decoherence, while the real story — dynamically and ontologically — is being told by the unfolding evolution of those branches. The “empty wave packets” in the configuration space which the corpuscles do not point at are none the worse for its absence: they still contain cells, dust motes, cats, people, wars and the like. The point has been stated clearly by Zeh:
 * It is usually overlooked that Bohm’s theory contains the same “many worlds” of dynamically separate branches as the Everett interpretation (now regarded as “empty” wave components), since it is based on precisely the same . . . global wave function . ..
 * Deutsch has expressed the point more acerbically:
 * [P]ilot-wave theories are parallel-universe theories in a state of chronic denial.

(page 12)
 * It is interesting that at times Bohmians slide into a mode of talking about systems as if they were just made out of corpuscles40, but this is only coherent if the radical position is adopted that the wavefunction is simply not real at all, that it is a piece of mathematical machinery in the quantum mechanical algorithm for the motion of corpuscles. Yet ‘reality’ is not some property which we can grant or withhold in an arbitrary way from the components of our mathematical formalism.

(page 13) Footnote 41:
 * Those who sympathise with Leibniz’ claim—fully endorsed by Einstein—that the essence of a real thing is its ability to act and be acted upon, may be interested in a defence of the reality of the wavefunction based on the action-reaction principle found in Anandan and Brown

(page 14/15) On Maudlin:
 * Why shouldn’t consciousness supervene as much on wavefunctions as on corpuscles?—a possibility that was clearly entertained by Bohm in 1951. But if one allows for this possibility, the floodgates into the Everettian multiplicity of autonomous, definite perceived outcomes are opened.[46] To restrict supervenience of consciousness to de Broglie-Bohm corpuscles in the brain does succeed in restricting conscious goings-on to one and only one branch of the Everett multiverse but it seems unwarranted and bizarre. The strategy seems unmotivated except by a desire purely to reduce the number of conscious observers in the universe, and it is at best unclear whether this is a reasonable application of Occam’s Razor.[47]

(page 15) Footnote 46:
 * As we saw in section 3, even in his hidden variables paper II of 1952, Bohm seems to associate the wavepacket chosen by the corpuscles as the representing outcome of the measurement—the role of the corpuscles merely being to point to it. But if this wavepacket can support consciousness, it is mysterious why empty ones cannot.

(page 15) Footnote 47:
 * It is noteworthy that the active role of the corpuscles in the de Broglie-Bohm theory is merely to act on each other, not back on the wavefunction. So it is striking that such passive entities are purportedly capable of grounding consciousness experience. C.f. Stone [11] and independently Brown [40].

(page 17) Parting words:
 * observation—in so far as this is related to the cognitive process of “knowing” the outcome of the measurement process—is not discovering the position of the de Broglie-Bohm corpuscle even if it exists.

Orange Marlin
Orange Marlin evidence

1st AN/I raised against MCP

2nd AN/I raised against MCP

I never accuse anyone of incivility

Hilarious protestations of innocence

I don't really care about {Civility}, and I never accuse anyone of incivility

Plants are not living organisms

Good and bad ghosts - is there a difference?
The article says:
 * The Faddeev-Popov ghosts are sometimes referred to as "good ghosts". The "bad ghosts" represent another, more general meaning of the word "ghost" in theoretical physics: states of negative norm - or fields with the wrong sign of the kinetic term - whose existence allows the probabilities to be negative.

But don't all ghosts imply negative norm states? I notice, reading Cheng and Li, that Faddeev-Popov ghost propagators always have the opposite sign from the analogous non-ghost propagators, which implies the opposite sign for their norm.--Michael C. Price talk 12:48, 2 July 2007 (UTC)

Dispute resolution
Resolving disputes Requests for arbitration Ignore all rules Policies and guidelines How to create policy Stable versions Criteria for speedy deletion Administrators' noticeboard/3RR

Please stop. If you continue to vandalize pages, you will be blocked from editing Wikipedia.

Complementarity critique

 * Niels Bohr stated "a complementary way of description is offered precisely by the quantum-mechanical formalism" (1949)


 * In this view, since the photons in the experiment obey the precise mathematical laws of quantum mechanics (the formalism), they can be described by Bohr's principle of complementarity. Cf:
 * "I think Bohr would have had no problem whatsoever with this experiment within his interpretation. Nor would any other interpretation of quantum mechanics. It is simply another manifestation of the admittedly strange, but utterly comprehensible (it can be calculated with exquisite precision), nature of quantum mechanics."
 * "There is absolutely nothing mysterious about Afshar's experiment. [....] And of course, the conventional quantum mechanics is compatible with the principle of complementarity."
 * "It was claimed that this experiment could be interpreted as a demonstration of a violation of the principle of complementarity in quantum mechanics. Instead, it is shown here that it can be understood in terms of classical wave optics and the standard interpretation of quantum mechanics."


 * Shahriar S. Afshar, "Sharp complementary wave and particle behaviours in the same welcher weg experiment", (2003) IRIMS www.irims.org/quant-ph/030503/; Proc. SPIE 5866 (2005) 229-244; AIP Cof. Proc. 810, (2006) 294-299. (Crossed beam experiment).


 * John G. Cramer, "A Farewell to Copenhagen?" (2005), Analog Science Fiction and Fact. (A non-technical discussion in a popular forum)

.
 * Marcus Chown,   "Afshar's Quantum Bombshell", "Quantum rebel" (July 24, 2004) New Scientist magazine; "A great leap forward",  (October 6, 2004) The Independent.
 * Ira Flatow, "Einstein, Bohr and the Nature of Light",, (July 30, 2004) Science Friday radio prgram, NPR.

History and Development
Orthomolecular megavitamin therapies, such as "megadose" usage of tocopherols and ascorbates, date back to the 1930s.

The term "orthomolecular" was first used by Linus Pauling in 1968, to express the "idea of the right molecules in the right amounts" and subsequently defined "orthomolecular medicine" as "the treatment of disease by the provision of the optimum molecular environment, especially the optimum concentrations of substances normally present in the human body." or as "the preservation of good health and the treatment of disease by varying the concentrations in the human body of substances that are normally present in the body and are required for health."

Since 1968 the orthomolecular field has developed further through the works of mainstream and non-mainstream researchers. Despite thus it still is often closely associated by the public with Pauling's advocacy of multi-gram doses of vitamin C for optimal health.

An example of a recent mainstream researcher is nutrition researcher Bruce Ames although he does not use the term itself. However his research deals with nutrition and specific genetic disease conditions (as indeed did Pauling's original article which defined the term "orthomolecular" ). Ames' research includes investigating the effects of large doses of, for example, the nutrients alpha-lipoic acid (a coenzyme precursor) and the carnitine (an amino acid complex) on restoring metabolic health, and in particular mitochondrial function, in animal models  Ames has also investigated the role of high dose B-vitamin therapy in alleviating in approximately 50 defective co-enzyme binding affinities, of which one, at least, every human suffers from (example of one genetic disease condition: Over 40% of the population is hetro- or homo-zygous with the thermolabile variant of 5,10-methylenetetrahydrofolate reductase and as a result requires extra riboflavin  ).

Ames has, based on his research, developed a supplement for human use.

Templates
this list of tags

Dirac equation
Before the cull

Definition and properties of the Landau-Lifshitz pseudotensor
The Landau-Lifshitz pseudotensor of the gravitational field has the following construction $$t_{LL}^{\mu \nu} = - \frac{1}{8\pi G}(G^{\mu \nu}) + \frac{1}{16\pi G (-g)}((-g)(g^{\mu \nu}g^{\alpha \beta} - g^{\mu \alpha}g^{\nu \beta})),_{\alpha \beta}$$

where:

$$G^{\mu \nu}$$ is the Einstein tensor

$$g^{\mu \nu}$$ is the metric tensor

$$g = \, \det \, (g_{\mu \nu})$$ is the determinant of a spacetime Lorentz metric

$$,_{\alpha \beta}$$ are partial derivatives, not covariant derivatives.

G is Newton's gravitational constant.

The Landau-Lifshitz pseudotensor is constructed so that when added to the stress-energy tensor of matter, $$T^{\mu \nu}$$, its total divergence vanishes:

$$((-g)(T^{\mu \nu} + t_{LL}^{\mu \nu})),{\mu} = 0 $$

This follows from the cancellation of the Einstein tensor, $$G^{\mu \nu}$$, with the stress-energy tensor, $$T^{\mu \nu}$$ by the Einstein field equations; the remaining term vanishes algebraically due the commutativity of partial derivatives applied across antisymmetric indices.

Aether critiques
Mainstream critics point out that Einstein's special theory of relativity is an extension of the principles of Galilean relativity or invariance from classical mechanics to include Maxwell's equations and thereby optics.

In the mainstream view, therefore, any attempt to formulate a new aether theory by recourse to Galilean relativity, is doomed since Galilean invariance is already incorporated into special relativity under the name Lorentz invariance; any putative aether is considered to be devoid of mechanical properties, unobservable and hence superfluous. It is held that any non-superfluous aether theory would yield predictions that are incompatible with Lorentz invariance and thereby Maxwell's equations; however the latter is empirically very well attested.

Consequently the concept of a "Galilean" aether or space has not been used in the Theory of Relativity, Quantum mechanics, or other modern theories of physics.

Michelson–Morley
This reference by Einstein in his 1905 paper is probably not about MMX, but to other attempts to detect the ether. Einstein is on record, early on, as saying that he hadn't heard of the MMX null result until after 1905, although later in his life, when we can presume his memory would not be so clear about distant events, he contradicted himself on this point. (Cf A P French's standard textbook (or see Michael Polanyi on this point) - French concludes that Einstein had not heard about the MMX -- and although you can find many texts that assume the reverse, they are wrong, IMO.)
 * Michael Polanyi, Personal Knowledge: Towards a Post-Critical Philosophy, ISBN: 0226672883, footnote page 10-11: Einstein reports, via Dr N Balzas in response to Polanyi's query, that "The Michelson-Morely experiment had no role in the foundation of the theory." and "..the theory of relativity was not founded to explain its outcome at all."

A P French, Special Relativity,
 * ISBN 0-442-30782-9 (Yellow cover)
 * ISBN 0-17-771075-6 (1968 /cheapest)

or
 * ISBN 9780393097931 (1968)

developed from? ISBN 0-393-09793-5 (1966) 0-412-34320-7

Matrices
$$\begin{matrix} |\mbox{electron, apparatus}_{before} \rang = |e^{-} \mbox{: x-spin} \rightarrow \rangle \otimes |\mbox{m: no bubbles} \rangle \\ \ =  (|e^{-} \mbox{: z-spin} \uparrow \rangle + |e^{-}\mbox{: z-spin} \downarrow \rangle)\frac{1}{\sqrt{2}}  \otimes |\mbox{m: no bubbles} \rangle \end{matrix}$$ $$\begin{matrix} = |e^{-}\mbox{: z-spin} \uparrow \rangle \otimes |\mbox{m: no bubbles} \rangle\frac{1}{\sqrt{2}}  + |e^{-}\mbox{: z-spin} \downarrow \rangle \otimes |\mbox{m: no bubbles} \rangle \frac{1}{\sqrt{2}} \end{matrix}$$

$$\begin{matrix} |\mbox{electron, apparatus}_{after} \rang =    \\ &      |e^{-}\mbox{: z-spin} \uparrow \rangle \otimes |\mbox{m: bubbles along upper path} \rangle \frac{1}{\sqrt{2}} \\ +&    |e^{-}\mbox{: z-spin} \downarrow \rangle \otimes |\mbox{m: bubbles along lower path} \rangle \frac{1}{\sqrt{2}} \end{matrix}$$

$$\begin{matrix} |\mbox{electron, apparatus}_{after} \rang = & \ \\ |e^{-}\mbox{: z-spin} \uparrow \rangle \otimes |\mbox{m: bubbles along upper path} \rangle\frac{1}{\sqrt{2}}   \  \end{matrix}$$ $$\begin{matrix} +  |e^{-}\mbox{: z-spin} \downarrow \rangle \otimes |\mbox{m: bubbles along lower path} \rangle \frac{1}{\sqrt{2}} \end{matrix}$$

Electroweak Lagrangian
The electroweak lagrangian can be written as :

$$\mathfrak{L}_{E-W} = \mathfrak{L}_g + \mathfrak{L}_f + \mathfrak{L}_H $$

The g term describes the gauge fields
 * $$\mathfrak{L}_g = -\tfrac{1}{4}G_a^{\mu\nu}G_{\mu\nu}^a - \tfrac{1}{4}B^{\mu\nu}B_{\mu\nu}$$

The f term describes the interaction between the electrons, muons, and quarks (the Dirac particles) of the Standard Model. The subscripts Li and Ri in $$\psi_{Li}$$ and $$\psi_{Ri}$$ refer to the Left and Right-handed spin of the i-th species of Dirac particles in the Standard Model. This is reflected in the asymmetric form of this term.
 * $$\mathfrak{L}_f = \sum_i \overline{\Psi_{\mathrm{L}i}}(i\delta\!\!\!/ - \tfrac{1}{2} gW\!\!\!\!\!/\;^at_a + \tfrac{1}{2}g'B\!\!\!\!/_y)\Psi_{\mathrm{L}i} + \sum_i \overline{\Psi_{\mathrm{R}i}}(i\delta\!\!\!/ + g'B\!\!\!\!/_y)\Psi_{\mathrm{R}i}$$

The H term describes the Higgs field $$\phi$$.
 * $$\mathfrak{L}_H = -(D_\nu\phi)^\mathsf{T}(D^\nu\phi) + \mu^2(\phi^\mathsf{T}\phi) - \lambda(\phi^\mathsf{T}\phi)^2$$

where
 * $$D_\nu\phi = $$

This gives rise to an effective lagrangian with a mass term, where the is mass generated by the interaction of the Higgs with the other varieties of particles given in the Lagrangian:
 * $$\mathfrak{L}_m = -\sum_{i,j} C_{ij}\overline{\Psi_{\mathrm{L}i}}\phi\Psi_{\mathrm{R}j}'$$

MWI measurement
Measurement and observation are easily handled in MWI. Measurements, or measurement-like interactions, are any interactions that correlate the observer's wavefunction with the observed system's wavefunction. A measurement, when the observed system is a definite state labelled by i, simply induces:
 * $$|O, i\rang => |O[i], i\rang$$

where O[i] represents the observer having detected the object system in the i-th state. In words this simply represents the observer measuring the observed system in the i-th state.

A measurement is complete when:
 * $$\lang O[i]|O[j]\rang = \delta_{ij}$$

Before the measurement has started the observer states are identical; after the measurement is complete the observer states are orthonormal. Thus a measurement defines the branching process: the branching is as well- or ill- defined as the measurement is. Thus branching is complete when the measurement is complete. Since the role of the observer and measurement per se plays no special role in MWI (measurements are handled as all other interactions are) there is no need for a precise definition of what an observer or a measurement is – just as in Newtonian physics no precise definition of either an observer or a measurement was required or expected. In all circumstances the universal wavefunction is still available to give a complete description of reality.

An illustrative example
MWI describes measurements as a formation of an entangled state which is a perfectly linear process (in terms of quantum superpositions) without any collapse of the wave function. For illustration, consider a Stern-Gerlach experiment and an electron or a silver atom passing this apparatus with a spin polarization in the left-right or x direction and thus a superposition of a spin up and a spin down state in up-down or z-direction. As a measuring apparatus, take a bubble or tracking chamber (a nonabsorbing particle detector). And finally let a cat observe the bubble tracks that form in the bubble chamber. The electron passes the apparatus and reach the same site in the end on either way so that, except for the up-down z-spin polarization, the state of the electron is finally the same regardless of the path taken (see The Feynman Lectures on Physics for a detailed discussion of such a setup). Before the measurement, the state of the electron and measuring apparatus is:

$$\begin{matrix} |\mbox{electron, apparatus}_{before} \rang = |e^{-} \mbox{: x-spin} \rightarrow \rangle \otimes |\mbox{m: no bubbles} \rangle \\ \ =  (|e^{-} \mbox{: z-spin} \uparrow \rangle + |e^{-}\mbox{: z-spin} \downarrow \rangle)\frac{1}{\sqrt{2}}  \otimes |\mbox{m: no bubbles} \rangle \end{matrix}$$ $$\begin{matrix} = |e^{-}\mbox{: z-spin} \uparrow \rangle \otimes |\mbox{m: no bubbles} \rangle\frac{1}{\sqrt{2}}  + |e^{-}\mbox{: z-spin} \downarrow \rangle \otimes |\mbox{m: no bubbles} \rangle \frac{1}{\sqrt{2}} \end{matrix}$$

The state is factorizable into a tensor factor for the electron and another factor for the measurement apparatus. After the spin measurement (bubble formation), the state is:

$$\begin{matrix} |\mbox{electron, apparatus}_{after} \rang =    \\ &      |e^{-}\mbox{: z-spin} \uparrow \rangle \otimes |\mbox{m: bubbles along upper path} \rangle \frac{1}{\sqrt{2}} \\ +&    |e^{-}\mbox{: z-spin} \downarrow \rangle \otimes |\mbox{m: bubbles along lower path} \rangle \frac{1}{\sqrt{2}} \end{matrix}$$

The state is no longer factorizable -- regardless of the vector basis chosen the state has to be expressed as the sum of a number of terms (in this example, at least two). The state of the above experiment is decomposed into a sum of two correlated or so-called entangled states ("worlds") both of which will have their individivual history without any further interaction or quantum interference between the two due to the physical linearity of quantum mechanics (the superposition principle): All processes in nature are linear and correspond to linear operators acting on each superposition component individually without any notice of the other components being present.

This would also be true for two non-entangled superposed states, but the latter can be detected by interference which is not possible for different entangled states (without reversing the entanglement first): Different entangled states cannot interfere; interactions with other systems will only result in a further entanglement of them as well. In the example above, the state of a Schrödinger cat watching the scene will be factorizable in the beginning (before watching)

but not in the end (after watching)

This example also shows that it's not the whole world that is split up into "many worlds", but only the part of the world that is entangled with the considered quantum event. This splitting tends to extend by interactions and can be visualised by a zipper or a DNA molecule which are in a similar way not completely opened instantaneously but opens gradually, element by element.

Imaginative readers will even see the zipper structure and the extending splitting in the formula:

If a system state is entangled with many other degrees of freedom (such as those in amplifiers, photographs, heat, sound, computer memory circuits, neurons, paper documents) in an experiment, this amounts to a thermodynamically irreversible process which is constituted of many small individually reversible processes at the atomic or subatomic level as is generally the case for thermodynamic irreversibility in classical or quantum statistical mechanics. Thus there is -- for thermodynamic reasons -- no way for an observer to completely reverse the entanglement and thus observe the other worlds by doing interference experiments on them. On the other hand, for small systems with few degrees of freedom this is feasible, as long as the investigated aspect of the system remains unentangled with the rest of the world.

The MWI thus solves the measurement problem of quantum mechanics by reducing measurements to cascades of entanglements.

The formation of an entangled state is a linear operation in terms of quantum superpositions. Consider for example the vector basis $$|e^{-}\mbox{: a}\rangle \otimes |\mbox{m: c}\rangle, |e^{-}\mbox{: a}\rangle \otimes |\mbox{m: d}\rangle, |e^{-}\mbox{: b}\rangle \otimes |\mbox{m: c}\rangle, |e^{-}\mbox{: b}\rangle \otimes |\mbox{m: d}\rangle$$

and the non-entangled initial state $$|\psi_1\rangle = |e^{-}\mbox{: a}\rangle \otimes |\mbox{m: c}\rangle$$

The linear (and unitary and thus reversible) operation (in terms of quantum superpositions) corresponding to the matrix

$$\begin{bmatrix} 0 & 0 & 1 & 0 \\ 1/\sqrt{2} & 1/\sqrt{2} & 0 & 0 \\ -1/\sqrt{2} & 1/\sqrt{2} & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

(in the above vector basis) will result in the entangled state $$|\psi_2\rangle = \frac{1}{\sqrt{2}} (|e^{-}\mbox{: a}\rangle \otimes |\mbox{m: d}\rangle - |e^{-}\mbox{: b}\rangle \otimes |\mbox{m: c}\rangle) $$