Quadrivettore Flusso del Numero di particelle: dialogo tra 2 studenti

Daneel Olivar si trovava nella biblioteca del dipartimento di fisica. Il pomeriggio era quasi giunto al suo termine e molti studenti se n’erano già andati, probabilmente per dirigersi verso la mensa dove cenare. Di fronte a lui, due tavoli più avanti, si trovava una studentessa, Susan Cooper, più avanti di lui nei corsi di un anno.

Daneel sapeva che Susan aveva superato piuttosto bene l’esame di Relatività Generale ma soprattutto che la sua passione per la fisica moderna era uguale se non superiore alla sua. Spesso aveva notato altri studenti andare a chiederle spiegazioni su argomenti che non avevano capito; i loro sguardi, dopo averle parlato, mostravano sempre un’evidente soddisfazione. E lui, che stava seguendo il corso di relatività generale con il professor Baley, aveva un dubbio a proposito del “Quadrivettore Flusso del Numero di particelle”, ma non poteva rivolgersi a lui, visto che sarebbe stato via per un giro di conferenze ancora per una settimana. Avrebbe potuto pazientare un po’ e al suo rientro chiedergli un incontro, ma non vedeva l’ora di scogliere quel dubbio che si frapponeva tra lui e l’argomento successivo, lo “Stress Energy Tensor”.

Così Daneel decise di rivolgersi a Susan, pur non conoscendola di persona. Si rese però conto, proprio quando stava per farsi avanti, di quanto lei fosse immersa in qualcosa di difficile. Poi, quando ormai non erano rimasti nella sala della biblioteca che loro due, mentre la vide alzarsi e dirigersi verso la lavagna bianca che si trovava in fondo alla sala, si fece coraggio e alzandosi dalla sedia disse: “Scusa, posso disturbarti un attimo? Sono uno studente del terzo anno. Mi chiamo Daneel Olivar”.

SUSAN – Ciao! Non mi disturbi affatto. Io mi chiamo Susan Cooper. Come posso esserti di aiuto?

DANEEL – Ho un dubbio a proposito del “Quadrivettore Flusso del Numero di particelle” e non trovo il modo di risolverlo. So che tu hai già studiato l’argomento l’anno scorso e volevo chiederti se potessi aiutarmi a venirne a capo.

SUSAN – Ok, ci posso provare. Ho giusto una curiosità prima di entrare nell’argomento. Perché io? La tua è una scelta casuale dettata solo dal sapere che ho già seguito il corso di Relatività Generale?

DANEEL – No Susan. Il fatto è che non ho potuto fare a meno di notare con quanta passione studi la fisica e quanti studenti si rivolgano a te per chiarire i propri dubbi.

SUSAN – Capisco… Allora Daneel, dimmi di che si tratta. Sono curiosa!

DANEEL – Il mio dubbio riguarda la generalizzazione della misura del flusso del numero di particelle attraverso una qualunque superficie. Ho compreso i casi “normali”, cioè quelli in cui le superfici in questione sono quelle classiche, cioè con una delle coordinate costanti, che poi corrispondo alle componenti del quadrivettore stesso in un sistema di riferimento inerziale, ma non capisco, se non in modo intuitivo, come si sia giunti alla sua generalizzazione utilizzando il prodotto scalare tra il quadrivettore stesso e il vettore unitario normale alla superficie.

SUSAN – Ok Daneel. Ti è chiara la generalizzazione del concetto di superficie attraverso l’uso di un campo scalare \phi, utilizzando l’equazione \phi(T, x, y, z) = costante? E successivamente attraverso la definizione dello “unit normal one-form”?

DANEEL – Credo di sì.

SUSAN – Prova allora a raccontarmelo, come se dovessi spiegarlo a qualcuno che ancora non conosce l’argomento.

Susan notò un’espressione lievemente perplessa sul volto di Daneel.

SUSAN – Daneel, non voglio farti un esame, ci mancherebbe! Mi serve capire qual è il tuo grado di comprensione di ciò che sta alla la base del problema che mi hai descritto.

A questo punto, Daneel prese un pennarello e si mise alla lavagna…

DANEEL – Capisco! Dunque, possiamo in generale definire una superficie S come il luogo dei punti nel nostro “manifold”, lo spazio-tempo, che soddisfano l’equazione \phi(T, x, y, z) = costante, come hai già ricordato tu. In effetti il flusso del numero di particelle attraverso, per esempio una superficie con x costante, è coperta da questa definizione: basta considerare un campo che dipenda solo dalla coordinata x. Analogamente per le altre coordinate.
Ora se si considera un punto P su tale superficie e il gradiente di \phi in P, cioè \widetilde{d\phi}, e se \vec{V} è un vettore in P tangente alla superficie S, allora sicuramente \widetilde{d\phi} (\vec{V}) = 0.

SUSAN – E per quale motivo dovrebbe essere necessariamente \widetilde{d\phi} (\vec{V}) = 0?

DANEEL – Dire che \vec{V} è tangente ad S in P, significa che esiste una curva \gamma appartenente ad S e passante per P che definisce \vec{V}. Nel senso che per qualunque campo f \in C^\infty(M):

\vec{V} (f) = (f \circ \gamma)'(\lambda_0), essendo \gamma( \lambda_0 ) = P

e siccome \phi è costante lungo tutta \gamma, ne consegue che \vec{V} (\phi) = 0.

Una impercettibile espressione di soddisfazione apparve sul volto della ragazza.

SUSAN – Ok Daneel! Vai pure avanti.

DANEEL – \widetilde{d\phi} (\vec{V}) = \vec{V} ( d\phi ) = 0 per quanto appena visto.

Se ora usiamo il tensore metrico di Lorentz e supponiamo di essere in un sistema di coordinate inerziale, in cui quindi il tensore si può rappresentare come \eta_{\alpha\beta} = \begin{bmatrix}-1 & 0 & 0 \\0 & 1 & 0\\0 & 0 & 1 \end{bmatrix}, possiamo derivare un vettore da \widetilde{d\phi} che chiameremo \vec{d\phi}, definito come segue in termini delle sue componenti rispetto alla base indotta in P per il sistema inerziale in oggetto: (-\Phi_0, \Phi_1, \Phi_2 ,\Phi_3), essendo \Phi_i le componenti di \widetilde{d\phi} rispetto alla base duale.

Fece una piccola pausa e si voltò verso Susan, che continuava a guardare la lavagna assorbita.

Quindi <\vec{d\phi} \bullet  \vec{V}> = 0, visto che <\vec{V_1} \bullet  \vec{V_2}> := \eta( \vec{V_1} ,  \vec{V_2}) e che \eta( \vec{d\phi}, \vec{V}) = \widetilde{d\phi} (\vec{V}).

Ora possiamo finalmente defire il “normal unit one-form” e il suo corrispondente “normal unit vector”:

\widetilde{n} := \frac{\widetilde{d\phi}}{|\widetilde{d\phi}|} e \hat{n} :=  \frac{\vec{d\phi}}{|\vec{d\phi}|}

dove |\widetilde{d\phi}| = |\vec{d\phi}| := \sqrt{|\eta( \vec{d\phi}, \vec{d\phi} )|}.

SUSAN – Ottimo Daneel! Sei stato chiarissimo…

DANEEL – Non ci vuole ora molto per verificare che, scelto \phi in modo da definire una superficie “classica”, cioè una di quelle con x_i = costante, <\vec{N} \bullet \hat{n}> ci dà proprio il flusso del numero di particelle attraverso la superficie unitaria definita da \hat{n}, dove \vec{N} è il quadrivettore flusso del numero di particelle.

SUSAN – Direi che ora ci siamo. Tornando alla tua domanda iniziale…

DANEEL – Mi chiedevo come si potesse verificare che il flusso del numero di particelle attraverso una qualunque superficie unitaria sia effettivamente dato da <\vec{N} \bullet \hat{n}>.

Fece una piccola pausa…

In pratica, per essere più concreti, mentre è relativamente facile dedurre il flusso attraverso superfici classiche, le cose si complicano parecchio quando si considerino superfici in cui spazio e tempo variano insieme.

SUSAN – Sono d’accordo con te, Daneel. Se cerchiamo di arrivarci con un ragionamento “classico” è molto dura. Tuttavia la formula che stiamo valutando, cioè <\vec{N} \bullet \hat{n}>, è basata interamente su tensori. Il prodotto scalare altro non è che l’applicazione del tensore metrico, e \vec{N} e \hat{n} sono due vettori.

DANEEL – E quindi è invariante rispetto ad un cambio di sistema di coordinate…

SUSAN – Esatto!

DANEEL – Adesso capisco. Data una superficie che localmente, nel punto in cui vogliamo valutare il flusso di particelle attraverso di essa, risulti attraversare contemporaneamente spazio e tempo rispetto al sistema di coordinate scelto, possiamo sceglierne un altro rispetto a cui la superficie, almeno in un intorno sufficientemente piccolo del nostro punto, sia “standard”, cioè con una delle 4 coordinate fisse.

SUSAN – Esatto. Scegliendo opportunamente la velocità del nuovo sistema di riferimento e con una opportuna rotazione degli assi spaziali rispetto al sistema originario possiamo sempre riportarci in un caso “standard”.

DANEEL – E in tale sistema vale senz’altro l’equazione Fl =<\vec{N} \bullet \hat{n}> dove con Fl intendo il flusso (del numero di particelle attraverso la superficie nel punto P). Ma siccome esso, Fl, non dipende dal sistema di coordinate e anche <\vec{N} \bullet \hat{n}> non vi dipende, la formula ha carattere generale, indipendente dal sistema di coordinate scelto.

SUSAN – Proprio così Daneel! Questa è la potenza dei Tensori, oggetti matematici meravigliosi.

Daneel rimase in silenzio per un po’. Sembrava ci fosse ancora qualcosa che non gli tornava.

SUSAN – Ho la sensazione che ci sia ancora qualcosa che non ti convince, vero?

DANEEL – Tutto torna da un punto di vista prettamente matematico, tuttavia…è sul significato fisico del flusso del numero di particelle quando la superficie considerata attraversa tempo e spazio, che ho un dubbio. Come ben spiegato nel volume di Bernard Schutz su cui sto studiando (“A first course in General Relativity”), un’interpretazione di carattere generale del flusso prevede di considerare il numero di “world line” associate alle particelle che attraversano una sezione unitaria della superficie, che nel nostro caso, essendo lo spazio-tempo quadridimensionale, corrisponde ad un volume unitario. Tanto è vero che con questa interpretazione la densità di particelle, cioè la prima componente del quadrivettore flusso, può essere interpretata in modo analogo al flusso attraverso una superficie strettamente spaziale (per esempio con x costante)…

SUSAN – Se non ho capito male, il tuo dubbio riguarda il modo in cui si debba calcolare “l’area” della superficie considerata nel caso ci trovassimo in un sistema di riferimento in cui essa attraversi spazio e tempo contemporaneamente, giusto Daneel?

DANEEL – Proprio così Susan! Mi stupisce come tu capisca al volo i problemi…ma forse è perché ci avevi già pensato…

SUSAN – In effetti sì…sono una che scava e non si accontenta fino a che tutto non torna…direi come te, Daneel.

I due si scambiarono un breve sguardo d’intesa.

Nel nostro caso, più che di area dovremmo parlare di volume, come già ricordavi tu. Ora immagina di traslare, ruotare e modificare la velocità del nostro sistema di riferimento iniziale in modo da ricondurci ad un flusso “standard”, in pratica una delle 4 componenti del quadrivettore flusso del numero di particelle. Scegliamo ora il volumetto per il flusso come un cubetto i cui spigoli rappresentativi siano quindi tra loro ortogonali e di lunghezza piccola a piacere. Quando parliamo di lunghezza o di ortogonalità tra vettori nello spazio tempo, dobbiamo usare il tensore metrico di Lorentz! Ed è questo il punto cruciale: usando tale tensore, come è naturale, il nostro cubetto si mantiene tale in qualunque altro riferimento, sia in termini di volume sia di ortogonalità tra i suoi spigoli rappresentativi. Grazie quindi al tensore metrico, possiamo scegliere il nostro cubetto di volume desiderato, in qualunque sistema di riferimento!

DANEEL – Fantastico Susan! Non ho più dubbi…grazie mille!

Spacetime Loops in Quantum Eraser experiment

When considering the relationship between time and quantum mechanics, particularly interesting is the quantum eraser proposed by Scully and Drühl. For an outline of the experiment, you can see the figure below.

fig. 1

The photon passing through the lens and going to the “interference screen” is called the signal photon, the twin photon (entangled) is called the idler photon.

Now suppose to move all detectors (A, B, C and D) and associated instruments to a very high distance, e.g. 10 light years. Suppose also that photon gun produces one photon at a time. Which figure do you expect to see on the screen after a lot of photons has been fired by the photon gun?  As well explained by Brian you will see a figure without interference, because half of the idler photons are detected by A or B detectors . But if you consider only the signal photons associated to the idler detected by C and D detectors, you will discover they have formed an interference figure on the screen. The incredible thing is that this happens independently of the distance of detectors from the BBO crystal, or, in other words, independently of the time when the idler photons are detected.

But now suppose that the scientists in the laboratory at 10 light years from photon gun, have the possibility to modify the detectors configuration in one of 2 ways as shown below in fig. 2 and fig. 3:

fig. 2

fig. 3

Now suppose also that before going away from Earth they have established with scientist of laboratory with photon gun that every 6 minutes photon gun fires a beam of photons one at a time, separated by half of a second and that every beam lasts for 5 minutes. Their clocks are perfectly synchronized and we suppose that gravity effects on time are negligible. So in the minute that separates 2 subsequent beams of photons, scientist in the laboratory with detectors can change the detectors configuration in one of the two shown configurations in fig. 2 and 3.

When the configuration is that shown in fig. 2 (detectors A and B), 10 years before no interference figure has formed on the screen in laboratory on the Earth, Viceversa when configuration is that shown in fig. 3 (detectors C and D), 10 years before interference figure must have formed on the screen! 

From now on we will indicate with A the laboratory on Earth and with B that placed at 10 light years with the detectors.

Amazing consequences.

The consequences of all this are incredible!

From the point of view of scientists in laboratory A, they are able to predict the behavior of scientists in laboratory B! If they now observe the formation of an interference figure, it means that in 10 years from now the detectors in laboratory B will be in the configuration of fig. 3, while if they do not observe an interference figure then they will necessarily be in the configuration shown in fig. 2..

Let’s now put ourselves in the shoes of the scientists of laboratory B: they can determine with their current behavior, switching detectors configuration from that of fig. 2 to that of fig. 3, the formation or not of the interference figure in laboratory A, 10 years before!

They could use this mechanism to communicate messages to laboratory A in the past, assigning, e.g. the value 0 in the absence of the interference figure and the value 1 in its presence!

This is really incredible, because it seems to violate the cause-effect relationship that wants the cause precedes its effect!

Paradox.

We modify the scheme of figure 1, so that in laboratory B there is not a photon detector but a system of mirrors positioned in such a way as to send back the idler photons coming from laboratory A. We also modify laboratory A so that the 2 possible detectors configuration (A+B or C+D) are placed in it on the trajectory of idler photons reflected by mirrors in laboratory B. So scientist in laboratory A can choose to observe idler photons through detectors in configuration A+B or C+D.

Now scientists from laboratory A can cause the interference pattern to form or not for each beam of photons emitted by the gun, by placing the associated detectors configuration 20 years later (see fig. 4)! But this puts them in a position to violate the consequences of quantum mechanics: in fact they can always decide to position the detector in the opposite way to what is required. For example, if at a given moment they observe an interference pattern, 20 years later they can decide to choose detectors configuration as shown in fig. 2, giving rise to a paradox (the figure they should have observed 20 years earlier, had to be without the typical pattern of interference).

fig. 4

There is another situation that causes the paradox. Look at figure 5 below.

fig. 5

In A1 the laboratory A produces a beam of signal photons and associated idler photon sent to laboratory B and received in B1, where the idler photons can be detected as shown in fig. 2 (A+B) or can be detected as shown in fig. 3 (C+D) through the quantum eraser configuration. The same thing happens between B2 and A2, with reversed parts.

When laboratory A is in A2, it knows which kind of figure formed in its interference screen (interference pattern or not) and so through A2-B2 connection it can “cause” the opposite figure in laboratory B at point B2. And so laboratory B can cause that figure in A1 through B2, where it detects the idler photons arriving from A1. So we can have again the contradiction!

An attempt at resolution.

In this chapter I suggest a solution to the paradox just seen, supposing we can avoid to consider gravity effects, and so treat only inertial reference systems related to each other through special relativity transformation laws (Lorentz transformations).

I want to consider 2 important phenomena of Physics: simultaneity and entanglement. We know that entanglement happens between 2 particles, started from the same origin in a certain instant and observed at the same time. So entanglement requires that we measure the same physical quantity of the 2 particles at the same instant!

But we know that simultaneity of 2 events is not absolute.

Consider the figure below, where the trajectories of 2 entangled particles emitted from a source positioned at the origin of the reference system shown, are represented in the Minkowsky space time.

fig. 6

Event A, in the trajectory of particle 2, can be in a simultaneity relationship with all events between B and C belonging to particle 1 trajectory, because for any event inside the interval BC (we can call it H), an opportune reference system can exist from whose point of view event A and H are simultaneous. 

In the case H is one of the 2 extremes (B and C), the reference system from whose point of view the 2 considered events are simultaneous, is that attached to the photon going from B to A or from A to C respectively.

If we measure particle 2 in A (obviously we are measuring a specific observable of particle 2) what happens to particle 1? Where does its wave function collapse between B and C? There is only one possibility: it must collapse in B and it cannot change from B to C, otherwise there would exist at least one reference system from whose point of view the 2 particles wouldn’t be entangled. We’ll come back on this later.

A new type of simultaneity

After these considerations we can introduce a new type of simultaneity, which I’ll call absolute simultaneity (abbreviated as a-simultaneity), defined as follow: 2 events are in an a-simultaneity relationship if an opportune inertial reference system can be imagined, with respect to which they are simultaneous.

With this definition, we infer that all events belonging to the trajectories of 2 photons starting from the same origin at the same time, are in an a-simultaneity relationship.

For simplicity, from now on, we’ll say that 2 events are a-simultaneous meaning that they are in a-simultaneity relationship.

Direct Paths

Now I want to investigate which are the simplest paths through which information can travel.

First we need to introduce the concept of “direct path” between 2 events A and B: it is the shortest path from A to B, extending the classic Euclidean distance of a 3 dimensional space, to the 4 dimensional space time, in which we define the 4th coordinate as T=ct (where t is the reference system time and c the speed of light). With this definition we can introduce 5 types of “direct path”:

  1. a path from past to future, abbreviated to PF
  2. a path from future to past, abbreviated to FP
  3. a path along the geodesic trajectory of a photon from past to future: abbreviated to PHPF
  4. a path along the geodesic  trajectory of a photon from future to past: abbreviated to PHFP
  5. a path connecting 2 a-simultaneous events, abbreviated to AS

Direct paths PHPF and PHFP are special cases of AS path, but we need to specify them as you will understand in a while. Moreover we use the past-to-future and future-to-past concepts to define them because, although they are a-simultaneous path, for any reference system excluded that of an hypothetical photon, they “move” from past to future and in the opposite direction respectively.

Below a representation of all possible direct paths.

fig. 7

Some of these are the basic types of path through which information can travel, others are forbidden paths.

The basic types of direct path through which Information can travel are:

  • PF: this is obvious;
  • PHPF: this is also obvious because we use electromagnetic signals to transmit information;
  • PHFP: this is the way information travels in the case of entanglement;

In all the other cases information cannot travel directly. 

In the case of AS path, it can travel through the composition of:

  • a PHFP with a PF
  • or a PHFP with a PHPF

The FP direct path for information is not admitted, but information can travel through the composition (for example) of 2 PHFP!

Now we introduce a new rule of Nature which is about loops of information, where a loop is a trajectory where information starts from a point (belonging to it) and through a sequence of basic direct paths returns to the starting point. 

It says: “No information loops in spacetime, formed by basic admitted direct paths, are possible”.

And now, with this new rule we can try to solve the 2 paradoxes introduced before and we can try to have a deeper view on entanglement.

Solving Paradox of fig. 4

Now we can also try to solve the paradox described through fig. 4, which is reproduced below:

First we need to understand where information starts. There only 2 possibilities: A or C. But we know that the figure formed on the screen in A, is a consequence of the detectors configuration in C: so C is where information starts! And now the question: does it travel through a loop made with allowed direct paths? Yes! The loop is formed by:

  1. C-B: which is a PHFP path
  2. B-A: which is a PHFP path too
  3. A-C: which is a PF path

But this violates the rule postulated about information loops. So the Paradox is not possible if we accept this rule.

Solving Paradox of fig. 5

For clarity the paradox is reproduced below (same as fig.5):

Where information starts? As for the previous case, it starts in A2, where scientists of laboratory A decide detectors configuration. And as before we see there is a loop:

  1. A2-B2: PHFP path
  2. B2-B1: PF path
  3. B1-A1: PHFP path
  4. A1-A2: PF path

And as before, if we accept the new rule about information loops, it cannot happen!

How Nature avoid loops?

What happens if there’s a loop as in the case of the 2 paradoxes? I think that simply Nature avoids the loop, breaking the connection between the idler photons and the associated signal photons, so that these always form an interference pattern on the screen. No information will go through the PHFP paths.

A more general case

We now consider the case illustrated in the figure below, where B-M and M-A are PHFP direct paths and A-B is a PF path which can be used by laboratory A to send information to laboratory B. 

fig. 8

Here the problem is that laboratory A could send information about the pattern formed on its screen by signal photons, or could not. This case includes the previous two, considering that laboratory A could forget information about the pattern formed by signal photons! Sending information in those cases corresponds to keep it.

How can Nature know if laboratory A sends information or not after having seen the pattern formed by the signal photons? Remember that the type of pattern formed by signal photons happens before the information is sent to B.

I think the only possibility is to relax the rule about information loops in this way: “No information loops in spacetime, formed by possible basic admitted direct paths, are possible”.

Possible means that it is really possible that information travels on that path. If the instruments of lab. A sending information are broken, Nature “knows” that there is no possibility that information reaches the lab. B on time. Quantum mechanics has taught us that possibilities are real. All possible evolutions of a system are real, even if only one is observed, When laboratory A sends the idler photons, Nature says: “is its state compatible with the possibility that it sends information to laboratory B?”. And if the possibility of that is not 0 then it must break the loop, always in the same way, breaking the correlation between idler and signal photons: only interference patterns will be observed on the screen in lab. A.

To be clearer we consider some examples.

Suppose that the device used in laboratory A to decide whether sending or not information to laboratory B about the pattern formed by the signal photons, is based on quantum mechanics: for example and head or tales quantum coin, practically a quantum bit with state 1/sqrt(2)|0> + 1/sqrt(2)|1>. If the observation is head then information will be sent, otherwise it will not. Nature cannot rules out the possibility that head will be the result, so it has to break the loop as explained before.

Suppose now that the device used for deciding whether sending or not information to laboratory B, is a classical stochastic head or tails coin (in practice a real coin). Because it “does not obey quantum laws”, in the sense that its behaviour is practically deterministic, Nature knows with precision which face of the coin will be the result, so only in the case that the head will be the result it will break the loop.

And what will happen if the “device” used to decide whether sending or not information is a human being? Because a human being can always decide to make fun of Nature (and its laws) the loop will be always broken.

Maybe this is proof that the human brain works (not only but also) as a quantum system!

On Entanglement

In the case of 2 entangled particles, which are measured at the same time with respect to an appropriate reference system, if we choose the observation of one of them as the starting point of information (although this information is not usable to send messages to anyone), we see that it travels on a loop. This seems to violate the rule about information loops in spacetime, but it’s not the case, because there is a symmetry (a reflection symmetry) between the 2 particles; there isn’t one source of information, both the observations of the 2 particles cause information travelling. There is not a loop of information but 2 symmetric paths starting from 2 symmetric origins.

About the speed of light.

Another interesting consideration concerns the speed of photons. We know that in special relativity the fundamental postulate is that the speed of light is constant and does not vary passing from one inertial reference to another. 

Let’s imagine for a moment that the scientists of the 2 laboratories know nothing of this postulate and suppose that the scientists of laboratory A wonder if it is possible to warn of the result of a sequence of photons the scientists of laboratory B before this is reached by the sequence of idler photons. If this were possible, the scientists of laboratory B could decide to position the detectors in the opposite configuration with respect to the figure that emerged on the screen of laboratory A, creating an evident contradiction in quantum mechanics. They know that quantum mechanics is correct and therefore deduce that it is not possible to transmit information that travels faster than the photons themselves. but still being tied to the classical vision of space and time, they imagine a stratagem: a spaceship at very high speed will pass in correspondence with laboratory A just when the sequence of photons formed the figure on the screen, and recovered this information it will transmit it to the laboratory B through an electromagnetic signal (photons); as they consider the Galilean transformations valid, they deduce that the photons emitted by the spaceship will have higher speeds than the idler ones coming from the laboratory, therefore laboratory B will be informed on the figure appeared on the screen before receiving the photons idler themselves. As before then the scientists of the lab. B could change detectors configuration in such a way to contradict what is foreseen by quantum mechanics. Once again they know that this is not possible, so they deduce that the photons emitted by the spacecraft cannot travel faster than those coming out of the lab. A. And since one would not understand why they should travel slower, they deduce that the speed of photons does not depend on the speed of their source. At the end they come to the conclusion that light must have the same speed for any observer!

It therefore seems possible to deduce the postulate on the constancy of the speed of light for any inertial reference, starting from quantum mechanics.